atlas of lie groups and representations · 2018. 10. 20. · atlas of lie groups and...
TRANSCRIPT
![Page 1: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/1.jpg)
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Atlas of Lie Groups and Representations
Jeffrey AdamsSun Yat-sen University, Guanhzhou
August 4, 2018
Slides available at: www.liegroups.org
![Page 3: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/3.jpg)
Atlas Project
Jeffrey Adams
Dan Barbasch
Birne Binegar
Fokko du Cloux
Marc van Leeuwen
Marc Noel
Annegret Paul
Susana Salamanca
Siddhartha Sahi
John Stembridge
Peter Trapa
David Vogan
![Page 4: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/4.jpg)
Overview
Atlas software:
Computations in Lie theory
Computing unitary representations, the unitary dual, many otherthings
Software is freely available www.liegroups.org
See www.liegroups.org/help
1) Learn by examples.
2) You only really understand something if you can program it on acomputer.
![Page 5: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/5.jpg)
Overview
Atlas software:
Computations in Lie theory
Computing unitary representations, the unitary dual, many otherthings
Software is freely available www.liegroups.org
See www.liegroups.org/help
1) Learn by examples.
2) You only really understand something if you can program it on acomputer.
![Page 6: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/6.jpg)
Overview
Atlas software:
Computations in Lie theory
Computing unitary representations, the unitary dual,
many otherthings
Software is freely available www.liegroups.org
See www.liegroups.org/help
1) Learn by examples.
2) You only really understand something if you can program it on acomputer.
![Page 7: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/7.jpg)
Overview
Atlas software:
Computations in Lie theory
Computing unitary representations, the unitary dual, many otherthings
Software is freely available www.liegroups.org
See www.liegroups.org/help
1) Learn by examples.
2) You only really understand something if you can program it on acomputer.
![Page 8: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/8.jpg)
Overview
Atlas software:
Computations in Lie theory
Computing unitary representations, the unitary dual, many otherthings
Software is freely available www.liegroups.org
See www.liegroups.org/help
1) Learn by examples.
2) You only really understand something if you can program it on acomputer.
![Page 9: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/9.jpg)
Overview
Atlas software:
Computations in Lie theory
Computing unitary representations, the unitary dual, many otherthings
Software is freely available www.liegroups.org
See www.liegroups.org/help
1) Learn by examples.
2) You only really understand something if you can program it on acomputer.
![Page 10: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/10.jpg)
Overview
Atlas software:
Computations in Lie theory
Computing unitary representations, the unitary dual, many otherthings
Software is freely available www.liegroups.org
See www.liegroups.org/help
1) Learn by examples.
2) You only really understand something if you can program it on acomputer.
![Page 11: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/11.jpg)
Overview
Atlas software:
Computations in Lie theory
Computing unitary representations, the unitary dual, many otherthings
Software is freely available www.liegroups.org
See www.liegroups.org/help
1) Learn by examples.
2) You only really understand something if you can program it on acomputer.
![Page 12: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/12.jpg)
Overview
Atlas software:
Computations in Lie theory
Computing unitary representations, the unitary dual, many otherthings
Software is freely available www.liegroups.org
See www.liegroups.org/help
1) Learn by examples.
2) You only really understand something if you can program it on acomputer.
![Page 13: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/13.jpg)
Before we begin. . .
Note: Everything is in the context of G (C) is a connected complexreductive group,
G (R) a real form of G (C)
K (R) ⊂ G (R) maximal compact, K (C) ⊂ G (C)
“forget about K (R)”:everything in terms of G (C),K (C):
Unless otherwise noted: (almost) everything in sight is complexand complex algebraic
![Page 14: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/14.jpg)
Before we begin. . .
Note: Everything is in the context of G (C) is a connected complexreductive group,
G (R) a real form of G (C)
K (R) ⊂ G (R) maximal compact, K (C) ⊂ G (C)
“forget about K (R)”:everything in terms of G (C),K (C):
Unless otherwise noted: (almost) everything in sight is complexand complex algebraic
![Page 15: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/15.jpg)
Before we begin. . .
Note: Everything is in the context of G (C) is a connected complexreductive group,
G (R) a real form of G (C)
K (R) ⊂ G (R) maximal compact, K (C) ⊂ G (C)
“forget about K (R)”:everything in terms of G (C),K (C):
Unless otherwise noted: (almost) everything in sight is complexand complex algebraic
![Page 16: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/16.jpg)
Before we begin. . .
Note: Everything is in the context of G (C) is a connected complexreductive group,
G (R) a real form of G (C)
K (R) ⊂ G (R) maximal compact, K (C) ⊂ G (C)
“forget about K (R)”:
everything in terms of G (C),K (C):
Unless otherwise noted: (almost) everything in sight is complexand complex algebraic
![Page 17: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/17.jpg)
Before we begin. . .
Note: Everything is in the context of G (C) is a connected complexreductive group,
G (R) a real form of G (C)
K (R) ⊂ G (R) maximal compact, K (C) ⊂ G (C)
“forget about K (R)”:everything in terms of G (C),K (C):
Unless otherwise noted: (almost) everything in sight is complexand complex algebraic
![Page 18: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/18.jpg)
Before we begin. . .
Note: Everything is in the context of G (C) is a connected complexreductive group,
G (R) a real form of G (C)
K (R) ⊂ G (R) maximal compact, K (C) ⊂ G (C)
“forget about K (R)”:everything in terms of G (C),K (C):
Unless otherwise noted: (almost) everything in sight is complexand complex algebraic
![Page 19: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/19.jpg)
Structure Theory
1. Root data and reductive groups
1.1 Root Data1.2 roots, coroots, simple roots, highest root,fundamental
weights. . .1.3 Weyl group1.4 Complex connected reductive groups1.5 Radical, center, derived group
![Page 20: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/20.jpg)
Structure Theory
1. Root data and reductive groups
1.1 Root Data
1.2 roots, coroots, simple roots, highest root,fundamentalweights. . .
1.3 Weyl group1.4 Complex connected reductive groups1.5 Radical, center, derived group
![Page 21: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/21.jpg)
Structure Theory
1. Root data and reductive groups
1.1 Root Data1.2 roots, coroots, simple roots, highest root,fundamental
weights. . .
1.3 Weyl group1.4 Complex connected reductive groups1.5 Radical, center, derived group
![Page 22: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/22.jpg)
Structure Theory
1. Root data and reductive groups
1.1 Root Data1.2 roots, coroots, simple roots, highest root,fundamental
weights. . .1.3 Weyl group
1.4 Complex connected reductive groups1.5 Radical, center, derived group
![Page 23: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/23.jpg)
Structure Theory
1. Root data and reductive groups
1.1 Root Data1.2 roots, coroots, simple roots, highest root,fundamental
weights. . .1.3 Weyl group1.4 Complex connected reductive groups
1.5 Radical, center, derived group
![Page 24: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/24.jpg)
Structure Theory
1. Root data and reductive groups
1.1 Root Data1.2 roots, coroots, simple roots, highest root,fundamental
weights. . .1.3 Weyl group1.4 Complex connected reductive groups1.5 Radical, center, derived group
![Page 25: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/25.jpg)
Real Groups
2 Involutions of reductive groups, real groups
2.1 Cartan involution2.2 1→ Int(G )→ Aut(G )→ Out(G )→ 12.3 Pinnings, splitting of the exact sequence2.4 Inner classes of real forms
Example:G = SL(n,C)
Out(G ) = Z2 = {1, γ}
Inner class of 1: {SU(p, q) | p + q = n}
Inner class of γ (γ(g) =t g−1):SL(n,R), also SL(n/2,H)
![Page 26: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/26.jpg)
Real Groups
2 Involutions of reductive groups, real groups
2.1 Cartan involution2.2 1→ Int(G )→ Aut(G )→ Out(G )→ 12.3 Pinnings, splitting of the exact sequence2.4 Inner classes of real forms
Example:G = SL(n,C)
Out(G ) = Z2 = {1, γ}
Inner class of 1: {SU(p, q) | p + q = n}
Inner class of γ (γ(g) =t g−1):SL(n,R), also SL(n/2,H)
![Page 27: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/27.jpg)
Real Groups
2 Involutions of reductive groups, real groups
2.1 Cartan involution
2.2 1→ Int(G )→ Aut(G )→ Out(G )→ 12.3 Pinnings, splitting of the exact sequence2.4 Inner classes of real forms
Example:G = SL(n,C)
Out(G ) = Z2 = {1, γ}
Inner class of 1: {SU(p, q) | p + q = n}
Inner class of γ (γ(g) =t g−1):SL(n,R), also SL(n/2,H)
![Page 28: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/28.jpg)
Real Groups
2 Involutions of reductive groups, real groups
2.1 Cartan involution2.2 1→ Int(G )→ Aut(G )→ Out(G )→ 1
2.3 Pinnings, splitting of the exact sequence2.4 Inner classes of real forms
Example:G = SL(n,C)
Out(G ) = Z2 = {1, γ}
Inner class of 1: {SU(p, q) | p + q = n}
Inner class of γ (γ(g) =t g−1):SL(n,R), also SL(n/2,H)
![Page 29: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/29.jpg)
Real Groups
2 Involutions of reductive groups, real groups
2.1 Cartan involution2.2 1→ Int(G )→ Aut(G )→ Out(G )→ 12.3 Pinnings, splitting of the exact sequence
2.4 Inner classes of real forms
Example:G = SL(n,C)
Out(G ) = Z2 = {1, γ}
Inner class of 1: {SU(p, q) | p + q = n}
Inner class of γ (γ(g) =t g−1):SL(n,R), also SL(n/2,H)
![Page 30: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/30.jpg)
Real Groups
2 Involutions of reductive groups, real groups
2.1 Cartan involution2.2 1→ Int(G )→ Aut(G )→ Out(G )→ 12.3 Pinnings, splitting of the exact sequence2.4 Inner classes of real forms
Example:G = SL(n,C)
Out(G ) = Z2 = {1, γ}
Inner class of 1: {SU(p, q) | p + q = n}
Inner class of γ (γ(g) =t g−1):SL(n,R), also SL(n/2,H)
![Page 31: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/31.jpg)
Real Groups
2 Involutions of reductive groups, real groups
2.1 Cartan involution2.2 1→ Int(G )→ Aut(G )→ Out(G )→ 12.3 Pinnings, splitting of the exact sequence2.4 Inner classes of real forms
Example:G = SL(n,C)
Out(G ) = Z2 = {1, γ}
Inner class of 1: {SU(p, q) | p + q = n}
Inner class of γ (γ(g) =t g−1):SL(n,R), also SL(n/2,H)
![Page 32: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/32.jpg)
Real Groups
2 Involutions of reductive groups, real groups
2.1 Cartan involution2.2 1→ Int(G )→ Aut(G )→ Out(G )→ 12.3 Pinnings, splitting of the exact sequence2.4 Inner classes of real forms
Example:G = SL(n,C)
Out(G ) = Z2 = {1, γ}
Inner class of 1: {SU(p, q) | p + q = n}
Inner class of γ (γ(g) =t g−1):SL(n,R), also SL(n/2,H)
![Page 33: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/33.jpg)
Real Groups
2 Involutions of reductive groups, real groups
2.1 Cartan involution2.2 1→ Int(G )→ Aut(G )→ Out(G )→ 12.3 Pinnings, splitting of the exact sequence2.4 Inner classes of real forms
Example:G = SL(n,C)
Out(G ) = Z2 = {1, γ}
Inner class of 1: {SU(p, q) | p + q = n}
Inner class of γ (γ(g) =t g−1):SL(n,R), also SL(n/2,H)
![Page 34: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/34.jpg)
Real Groups
2 Involutions of reductive groups, real groups
2.1 Cartan involution2.2 1→ Int(G )→ Aut(G )→ Out(G )→ 12.3 Pinnings, splitting of the exact sequence2.4 Inner classes of real forms
Example:G = SL(n,C)
Out(G ) = Z2 = {1, γ}
Inner class of 1: {SU(p, q) | p + q = n}
Inner class of γ (γ(g) =t g−1):SL(n,R),
also SL(n/2,H)
![Page 35: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/35.jpg)
Real Groups
2 Involutions of reductive groups, real groups
2.1 Cartan involution2.2 1→ Int(G )→ Aut(G )→ Out(G )→ 12.3 Pinnings, splitting of the exact sequence2.4 Inner classes of real forms
Example:G = SL(n,C)
Out(G ) = Z2 = {1, γ}
Inner class of 1: {SU(p, q) | p + q = n}
Inner class of γ (γ(g) =t g−1):SL(n,R), also SL(n/2,H)
![Page 36: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/36.jpg)
K
3 Maximal compact subgroup K (R)
3.1 Maximal compact subgroup3.2 K (C) = G θ, K (R) compact3.3 Component groups: K/K 0 = K (R)/K (R)0 = G (R)/G (R)0
3.4 K orbits on G/B (B=Borel subgroup)
Example: G = GL(n,C)
θ(g) = Jp,qgJ−1p,q ,
Jp,q = diag(Ip,−Iq),K = GL(p,C)× GL(q,C),
K (R) = U(p)× U(q), G (R) = U(p, q)
θ(g) =t g−1, K (C) = O(n,C), K (R) = O(n), G (R) = GL(n,R)
Theorem: G/B = G (C)/B(C) is a projective variety, with finitelymany K = K (C)-orbits
3.5 Real Cartan subgroups, (relative) Weyl groups
![Page 37: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/37.jpg)
K
3 Maximal compact subgroup K (R)
3.1 Maximal compact subgroup3.2 K (C) = G θ, K (R) compact3.3 Component groups: K/K 0 = K (R)/K (R)0 = G (R)/G (R)0
3.4 K orbits on G/B (B=Borel subgroup)
Example: G = GL(n,C)
θ(g) = Jp,qgJ−1p,q ,
Jp,q = diag(Ip,−Iq),K = GL(p,C)× GL(q,C),
K (R) = U(p)× U(q), G (R) = U(p, q)
θ(g) =t g−1, K (C) = O(n,C), K (R) = O(n), G (R) = GL(n,R)
Theorem: G/B = G (C)/B(C) is a projective variety, with finitelymany K = K (C)-orbits
3.5 Real Cartan subgroups, (relative) Weyl groups
![Page 38: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/38.jpg)
K
3 Maximal compact subgroup K (R)
3.1 Maximal compact subgroup
3.2 K (C) = G θ, K (R) compact3.3 Component groups: K/K 0 = K (R)/K (R)0 = G (R)/G (R)0
3.4 K orbits on G/B (B=Borel subgroup)
Example: G = GL(n,C)
θ(g) = Jp,qgJ−1p,q ,
Jp,q = diag(Ip,−Iq),K = GL(p,C)× GL(q,C),
K (R) = U(p)× U(q), G (R) = U(p, q)
θ(g) =t g−1, K (C) = O(n,C), K (R) = O(n), G (R) = GL(n,R)
Theorem: G/B = G (C)/B(C) is a projective variety, with finitelymany K = K (C)-orbits
3.5 Real Cartan subgroups, (relative) Weyl groups
![Page 39: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/39.jpg)
K
3 Maximal compact subgroup K (R)
3.1 Maximal compact subgroup3.2 K (C) = G θ, K (R) compact
3.3 Component groups: K/K 0 = K (R)/K (R)0 = G (R)/G (R)0
3.4 K orbits on G/B (B=Borel subgroup)
Example: G = GL(n,C)
θ(g) = Jp,qgJ−1p,q ,
Jp,q = diag(Ip,−Iq),K = GL(p,C)× GL(q,C),
K (R) = U(p)× U(q), G (R) = U(p, q)
θ(g) =t g−1, K (C) = O(n,C), K (R) = O(n), G (R) = GL(n,R)
Theorem: G/B = G (C)/B(C) is a projective variety, with finitelymany K = K (C)-orbits
3.5 Real Cartan subgroups, (relative) Weyl groups
![Page 40: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/40.jpg)
K
3 Maximal compact subgroup K (R)
3.1 Maximal compact subgroup3.2 K (C) = G θ, K (R) compact3.3 Component groups: K/K 0 = K (R)/K (R)0 = G (R)/G (R)0
3.4 K orbits on G/B (B=Borel subgroup)
Example: G = GL(n,C)
θ(g) = Jp,qgJ−1p,q ,
Jp,q = diag(Ip,−Iq),K = GL(p,C)× GL(q,C),
K (R) = U(p)× U(q), G (R) = U(p, q)
θ(g) =t g−1, K (C) = O(n,C), K (R) = O(n), G (R) = GL(n,R)
Theorem: G/B = G (C)/B(C) is a projective variety, with finitelymany K = K (C)-orbits
3.5 Real Cartan subgroups, (relative) Weyl groups
![Page 41: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/41.jpg)
K
3 Maximal compact subgroup K (R)
3.1 Maximal compact subgroup3.2 K (C) = G θ, K (R) compact3.3 Component groups: K/K 0 = K (R)/K (R)0 = G (R)/G (R)0
3.4 K orbits on G/B (B=Borel subgroup)
Example: G = GL(n,C)
θ(g) = Jp,qgJ−1p,q ,
Jp,q = diag(Ip,−Iq),K = GL(p,C)× GL(q,C),
K (R) = U(p)× U(q), G (R) = U(p, q)
θ(g) =t g−1, K (C) = O(n,C), K (R) = O(n), G (R) = GL(n,R)
Theorem: G/B = G (C)/B(C) is a projective variety, with finitelymany K = K (C)-orbits
3.5 Real Cartan subgroups, (relative) Weyl groups
![Page 42: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/42.jpg)
K
3 Maximal compact subgroup K (R)
3.1 Maximal compact subgroup3.2 K (C) = G θ, K (R) compact3.3 Component groups: K/K 0 = K (R)/K (R)0 = G (R)/G (R)0
3.4 K orbits on G/B (B=Borel subgroup)
Example: G = GL(n,C)
θ(g) = Jp,qgJ−1p,q ,
Jp,q = diag(Ip,−Iq),K = GL(p,C)× GL(q,C),
K (R) = U(p)× U(q), G (R) = U(p, q)
θ(g) =t g−1, K (C) = O(n,C), K (R) = O(n), G (R) = GL(n,R)
Theorem: G/B = G (C)/B(C) is a projective variety, with finitelymany K = K (C)-orbits
3.5 Real Cartan subgroups, (relative) Weyl groups
![Page 43: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/43.jpg)
K
3 Maximal compact subgroup K (R)
3.1 Maximal compact subgroup3.2 K (C) = G θ, K (R) compact3.3 Component groups: K/K 0 = K (R)/K (R)0 = G (R)/G (R)0
3.4 K orbits on G/B (B=Borel subgroup)
Example: G = GL(n,C)
θ(g) = Jp,qgJ−1p,q ,
Jp,q = diag(Ip,−Iq),K = GL(p,C)× GL(q,C),
K (R) = U(p)× U(q), G (R) = U(p, q)
θ(g) =t g−1, K (C) = O(n,C), K (R) = O(n), G (R) = GL(n,R)
Theorem: G/B = G (C)/B(C) is a projective variety, with finitelymany K = K (C)-orbits
3.5 Real Cartan subgroups, (relative) Weyl groups
![Page 44: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/44.jpg)
K
3 Maximal compact subgroup K (R)
3.1 Maximal compact subgroup3.2 K (C) = G θ, K (R) compact3.3 Component groups: K/K 0 = K (R)/K (R)0 = G (R)/G (R)0
3.4 K orbits on G/B (B=Borel subgroup)
Example: G = GL(n,C)
θ(g) = Jp,qgJ−1p,q ,
Jp,q = diag(Ip,−Iq),
K = GL(p,C)× GL(q,C),
K (R) = U(p)× U(q), G (R) = U(p, q)
θ(g) =t g−1, K (C) = O(n,C), K (R) = O(n), G (R) = GL(n,R)
Theorem: G/B = G (C)/B(C) is a projective variety, with finitelymany K = K (C)-orbits
3.5 Real Cartan subgroups, (relative) Weyl groups
![Page 45: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/45.jpg)
K
3 Maximal compact subgroup K (R)
3.1 Maximal compact subgroup3.2 K (C) = G θ, K (R) compact3.3 Component groups: K/K 0 = K (R)/K (R)0 = G (R)/G (R)0
3.4 K orbits on G/B (B=Borel subgroup)
Example: G = GL(n,C)
θ(g) = Jp,qgJ−1p,q ,
Jp,q = diag(Ip,−Iq),K = GL(p,C)× GL(q,C),
K (R) = U(p)× U(q), G (R) = U(p, q)
θ(g) =t g−1, K (C) = O(n,C), K (R) = O(n), G (R) = GL(n,R)
Theorem: G/B = G (C)/B(C) is a projective variety, with finitelymany K = K (C)-orbits
3.5 Real Cartan subgroups, (relative) Weyl groups
![Page 46: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/46.jpg)
K
3 Maximal compact subgroup K (R)
3.1 Maximal compact subgroup3.2 K (C) = G θ, K (R) compact3.3 Component groups: K/K 0 = K (R)/K (R)0 = G (R)/G (R)0
3.4 K orbits on G/B (B=Borel subgroup)
Example: G = GL(n,C)
θ(g) = Jp,qgJ−1p,q ,
Jp,q = diag(Ip,−Iq),K = GL(p,C)× GL(q,C),
K (R) = U(p)× U(q)
, G (R) = U(p, q)
θ(g) =t g−1, K (C) = O(n,C), K (R) = O(n), G (R) = GL(n,R)
Theorem: G/B = G (C)/B(C) is a projective variety, with finitelymany K = K (C)-orbits
3.5 Real Cartan subgroups, (relative) Weyl groups
![Page 47: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/47.jpg)
K
3 Maximal compact subgroup K (R)
3.1 Maximal compact subgroup3.2 K (C) = G θ, K (R) compact3.3 Component groups: K/K 0 = K (R)/K (R)0 = G (R)/G (R)0
3.4 K orbits on G/B (B=Borel subgroup)
Example: G = GL(n,C)
θ(g) = Jp,qgJ−1p,q ,
Jp,q = diag(Ip,−Iq),K = GL(p,C)× GL(q,C),
K (R) = U(p)× U(q), G (R) = U(p, q)
θ(g) =t g−1, K (C) = O(n,C), K (R) = O(n), G (R) = GL(n,R)
Theorem: G/B = G (C)/B(C) is a projective variety, with finitelymany K = K (C)-orbits
3.5 Real Cartan subgroups, (relative) Weyl groups
![Page 48: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/48.jpg)
K
3 Maximal compact subgroup K (R)
3.1 Maximal compact subgroup3.2 K (C) = G θ, K (R) compact3.3 Component groups: K/K 0 = K (R)/K (R)0 = G (R)/G (R)0
3.4 K orbits on G/B (B=Borel subgroup)
Example: G = GL(n,C)
θ(g) = Jp,qgJ−1p,q ,
Jp,q = diag(Ip,−Iq),K = GL(p,C)× GL(q,C),
K (R) = U(p)× U(q), G (R) = U(p, q)
θ(g) =t g−1,
K (C) = O(n,C), K (R) = O(n), G (R) = GL(n,R)
Theorem: G/B = G (C)/B(C) is a projective variety, with finitelymany K = K (C)-orbits
3.5 Real Cartan subgroups, (relative) Weyl groups
![Page 49: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/49.jpg)
K
3 Maximal compact subgroup K (R)
3.1 Maximal compact subgroup3.2 K (C) = G θ, K (R) compact3.3 Component groups: K/K 0 = K (R)/K (R)0 = G (R)/G (R)0
3.4 K orbits on G/B (B=Borel subgroup)
Example: G = GL(n,C)
θ(g) = Jp,qgJ−1p,q ,
Jp,q = diag(Ip,−Iq),K = GL(p,C)× GL(q,C),
K (R) = U(p)× U(q), G (R) = U(p, q)
θ(g) =t g−1, K (C) = O(n,C),
K (R) = O(n), G (R) = GL(n,R)
Theorem: G/B = G (C)/B(C) is a projective variety, with finitelymany K = K (C)-orbits
3.5 Real Cartan subgroups, (relative) Weyl groups
![Page 50: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/50.jpg)
K
3 Maximal compact subgroup K (R)
3.1 Maximal compact subgroup3.2 K (C) = G θ, K (R) compact3.3 Component groups: K/K 0 = K (R)/K (R)0 = G (R)/G (R)0
3.4 K orbits on G/B (B=Borel subgroup)
Example: G = GL(n,C)
θ(g) = Jp,qgJ−1p,q ,
Jp,q = diag(Ip,−Iq),K = GL(p,C)× GL(q,C),
K (R) = U(p)× U(q), G (R) = U(p, q)
θ(g) =t g−1, K (C) = O(n,C), K (R) = O(n),
G (R) = GL(n,R)
Theorem: G/B = G (C)/B(C) is a projective variety, with finitelymany K = K (C)-orbits
3.5 Real Cartan subgroups, (relative) Weyl groups
![Page 51: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/51.jpg)
K
3 Maximal compact subgroup K (R)
3.1 Maximal compact subgroup3.2 K (C) = G θ, K (R) compact3.3 Component groups: K/K 0 = K (R)/K (R)0 = G (R)/G (R)0
3.4 K orbits on G/B (B=Borel subgroup)
Example: G = GL(n,C)
θ(g) = Jp,qgJ−1p,q ,
Jp,q = diag(Ip,−Iq),K = GL(p,C)× GL(q,C),
K (R) = U(p)× U(q), G (R) = U(p, q)
θ(g) =t g−1, K (C) = O(n,C), K (R) = O(n), G (R) = GL(n,R)
Theorem: G/B = G (C)/B(C) is a projective variety, with finitelymany K = K (C)-orbits
3.5 Real Cartan subgroups, (relative) Weyl groups
![Page 52: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/52.jpg)
K
3 Maximal compact subgroup K (R)
3.1 Maximal compact subgroup3.2 K (C) = G θ, K (R) compact3.3 Component groups: K/K 0 = K (R)/K (R)0 = G (R)/G (R)0
3.4 K orbits on G/B (B=Borel subgroup)
Example: G = GL(n,C)
θ(g) = Jp,qgJ−1p,q ,
Jp,q = diag(Ip,−Iq),K = GL(p,C)× GL(q,C),
K (R) = U(p)× U(q), G (R) = U(p, q)
θ(g) =t g−1, K (C) = O(n,C), K (R) = O(n), G (R) = GL(n,R)
Theorem: G/B = G (C)/B(C) is a projective variety, with finitelymany K = K (C)-orbits
3.5 Real Cartan subgroups, (relative) Weyl groups
![Page 53: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/53.jpg)
K
3 Maximal compact subgroup K (R)
3.1 Maximal compact subgroup3.2 K (C) = G θ, K (R) compact3.3 Component groups: K/K 0 = K (R)/K (R)0 = G (R)/G (R)0
3.4 K orbits on G/B (B=Borel subgroup)
Example: G = GL(n,C)
θ(g) = Jp,qgJ−1p,q ,
Jp,q = diag(Ip,−Iq),K = GL(p,C)× GL(q,C),
K (R) = U(p)× U(q), G (R) = U(p, q)
θ(g) =t g−1, K (C) = O(n,C), K (R) = O(n), G (R) = GL(n,R)
Theorem: G/B = G (C)/B(C) is a projective variety, with finitelymany K = K (C)-orbits
3.5 Real Cartan subgroups, (relative) Weyl groups
![Page 54: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/54.jpg)
Example: SO(4,4)
![Page 55: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/55.jpg)
Langlands classification
4 Langlands Classification
4.1 Parameters for representations4.2 (g,K )-modules, Note: G = G (C),K = K (C)4.3 Admissible representations4.4 Infinitesimal character4.5 Mγ = {γ1, . . . , γn} (parameters for irreducible representation with
infinitesimal character γ)Harish-Chandra: finite set
4.6 Cuspidal data: π = IndG
P (πM)
4.7 theta-stable data: π = RGq (πL)
![Page 56: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/56.jpg)
Langlands classification
4 Langlands Classification
4.1 Parameters for representations4.2 (g,K )-modules, Note: G = G (C),K = K (C)4.3 Admissible representations4.4 Infinitesimal character4.5 Mγ = {γ1, . . . , γn} (parameters for irreducible representation with
infinitesimal character γ)Harish-Chandra: finite set
4.6 Cuspidal data: π = IndG
P (πM)
4.7 theta-stable data: π = RGq (πL)
![Page 57: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/57.jpg)
Langlands classification
4 Langlands Classification
4.1 Parameters for representations
4.2 (g,K )-modules, Note: G = G (C),K = K (C)4.3 Admissible representations4.4 Infinitesimal character4.5 Mγ = {γ1, . . . , γn} (parameters for irreducible representation with
infinitesimal character γ)Harish-Chandra: finite set
4.6 Cuspidal data: π = IndG
P (πM)
4.7 theta-stable data: π = RGq (πL)
![Page 58: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/58.jpg)
Langlands classification
4 Langlands Classification
4.1 Parameters for representations4.2 (g,K )-modules
, Note: G = G (C),K = K (C)4.3 Admissible representations4.4 Infinitesimal character4.5 Mγ = {γ1, . . . , γn} (parameters for irreducible representation with
infinitesimal character γ)Harish-Chandra: finite set
4.6 Cuspidal data: π = IndG
P (πM)
4.7 theta-stable data: π = RGq (πL)
![Page 59: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/59.jpg)
Langlands classification
4 Langlands Classification
4.1 Parameters for representations4.2 (g,K )-modules, Note: G = G (C),K = K (C)
4.3 Admissible representations4.4 Infinitesimal character4.5 Mγ = {γ1, . . . , γn} (parameters for irreducible representation with
infinitesimal character γ)Harish-Chandra: finite set
4.6 Cuspidal data: π = IndG
P (πM)
4.7 theta-stable data: π = RGq (πL)
![Page 60: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/60.jpg)
Langlands classification
4 Langlands Classification
4.1 Parameters for representations4.2 (g,K )-modules, Note: G = G (C),K = K (C)4.3 Admissible representations
4.4 Infinitesimal character4.5 Mγ = {γ1, . . . , γn} (parameters for irreducible representation with
infinitesimal character γ)Harish-Chandra: finite set
4.6 Cuspidal data: π = IndG
P (πM)
4.7 theta-stable data: π = RGq (πL)
![Page 61: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/61.jpg)
Langlands classification
4 Langlands Classification
4.1 Parameters for representations4.2 (g,K )-modules, Note: G = G (C),K = K (C)4.3 Admissible representations4.4 Infinitesimal character
4.5 Mγ = {γ1, . . . , γn} (parameters for irreducible representation withinfinitesimal character γ)Harish-Chandra: finite set
4.6 Cuspidal data: π = IndG
P (πM)
4.7 theta-stable data: π = RGq (πL)
![Page 62: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/62.jpg)
Langlands classification
4 Langlands Classification
4.1 Parameters for representations4.2 (g,K )-modules, Note: G = G (C),K = K (C)4.3 Admissible representations4.4 Infinitesimal character4.5 Mγ = {γ1, . . . , γn} (parameters for irreducible representation with
infinitesimal character γ)
Harish-Chandra: finite set
4.6 Cuspidal data: π = IndG
P (πM)
4.7 theta-stable data: π = RGq (πL)
![Page 63: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/63.jpg)
Langlands classification
4 Langlands Classification
4.1 Parameters for representations4.2 (g,K )-modules, Note: G = G (C),K = K (C)4.3 Admissible representations4.4 Infinitesimal character4.5 Mγ = {γ1, . . . , γn} (parameters for irreducible representation with
infinitesimal character γ)Harish-Chandra: finite set
4.6 Cuspidal data: π = IndG
P (πM)
4.7 theta-stable data: π = RGq (πL)
![Page 64: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/64.jpg)
Langlands classification
4 Langlands Classification
4.1 Parameters for representations4.2 (g,K )-modules, Note: G = G (C),K = K (C)4.3 Admissible representations4.4 Infinitesimal character4.5 Mγ = {γ1, . . . , γn} (parameters for irreducible representation with
infinitesimal character γ)Harish-Chandra: finite set
4.6 Cuspidal data: π = IndG
P (πM)
4.7 theta-stable data: π = RGq (πL)
![Page 65: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/65.jpg)
Langlands classification
4 Langlands Classification
4.1 Parameters for representations4.2 (g,K )-modules, Note: G = G (C),K = K (C)4.3 Admissible representations4.4 Infinitesimal character4.5 Mγ = {γ1, . . . , γn} (parameters for irreducible representation with
infinitesimal character γ)Harish-Chandra: finite set
4.6 Cuspidal data: π = IndG
P (πM)
4.7 theta-stable data: π = RGq (πL)
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Kazhdan-Lusztig-Vogan Polynomials
5 Kazhdan-Lusztig-Vogan Polynomials
5.1 Composition series: I (Γ) =∑
mΞ,ΓJ(Ξ)5.2 Character formula: J(Γ) =
∑MΞ,ΓI (Ξ)
5.3 PΞ,Γ ∈ Z[q]5.4 QΞ,Γ ∈ Z[q]5.5 mΞ,Γ = QΞ,Γ(q = 1)5.6 MΞ,Γ = ±PΞ,Γ(q = 1)5.7 Global character (function on regular semisimple set)5.8 (Twisted KLV polynomials)
![Page 67: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/67.jpg)
Kazhdan-Lusztig-Vogan Polynomials
5 Kazhdan-Lusztig-Vogan Polynomials
5.1 Composition series: I (Γ) =∑
mΞ,ΓJ(Ξ)5.2 Character formula: J(Γ) =
∑MΞ,ΓI (Ξ)
5.3 PΞ,Γ ∈ Z[q]5.4 QΞ,Γ ∈ Z[q]5.5 mΞ,Γ = QΞ,Γ(q = 1)5.6 MΞ,Γ = ±PΞ,Γ(q = 1)5.7 Global character (function on regular semisimple set)5.8 (Twisted KLV polynomials)
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Kazhdan-Lusztig-Vogan Polynomials
5 Kazhdan-Lusztig-Vogan Polynomials
5.1 Composition series: I (Γ) =∑
mΞ,ΓJ(Ξ)
5.2 Character formula: J(Γ) =∑
MΞ,ΓI (Ξ)5.3 PΞ,Γ ∈ Z[q]5.4 QΞ,Γ ∈ Z[q]5.5 mΞ,Γ = QΞ,Γ(q = 1)5.6 MΞ,Γ = ±PΞ,Γ(q = 1)5.7 Global character (function on regular semisimple set)5.8 (Twisted KLV polynomials)
![Page 69: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/69.jpg)
Kazhdan-Lusztig-Vogan Polynomials
5 Kazhdan-Lusztig-Vogan Polynomials
5.1 Composition series: I (Γ) =∑
mΞ,ΓJ(Ξ)5.2 Character formula: J(Γ) =
∑MΞ,ΓI (Ξ)
5.3 PΞ,Γ ∈ Z[q]5.4 QΞ,Γ ∈ Z[q]5.5 mΞ,Γ = QΞ,Γ(q = 1)5.6 MΞ,Γ = ±PΞ,Γ(q = 1)5.7 Global character (function on regular semisimple set)5.8 (Twisted KLV polynomials)
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Kazhdan-Lusztig-Vogan Polynomials
5 Kazhdan-Lusztig-Vogan Polynomials
5.1 Composition series: I (Γ) =∑
mΞ,ΓJ(Ξ)5.2 Character formula: J(Γ) =
∑MΞ,ΓI (Ξ)
5.3 PΞ,Γ ∈ Z[q]
5.4 QΞ,Γ ∈ Z[q]5.5 mΞ,Γ = QΞ,Γ(q = 1)5.6 MΞ,Γ = ±PΞ,Γ(q = 1)5.7 Global character (function on regular semisimple set)5.8 (Twisted KLV polynomials)
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Kazhdan-Lusztig-Vogan Polynomials
5 Kazhdan-Lusztig-Vogan Polynomials
5.1 Composition series: I (Γ) =∑
mΞ,ΓJ(Ξ)5.2 Character formula: J(Γ) =
∑MΞ,ΓI (Ξ)
5.3 PΞ,Γ ∈ Z[q]5.4 QΞ,Γ ∈ Z[q]
5.5 mΞ,Γ = QΞ,Γ(q = 1)5.6 MΞ,Γ = ±PΞ,Γ(q = 1)5.7 Global character (function on regular semisimple set)5.8 (Twisted KLV polynomials)
![Page 72: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/72.jpg)
Kazhdan-Lusztig-Vogan Polynomials
5 Kazhdan-Lusztig-Vogan Polynomials
5.1 Composition series: I (Γ) =∑
mΞ,ΓJ(Ξ)5.2 Character formula: J(Γ) =
∑MΞ,ΓI (Ξ)
5.3 PΞ,Γ ∈ Z[q]5.4 QΞ,Γ ∈ Z[q]5.5 mΞ,Γ = QΞ,Γ(q = 1)
5.6 MΞ,Γ = ±PΞ,Γ(q = 1)5.7 Global character (function on regular semisimple set)5.8 (Twisted KLV polynomials)
![Page 73: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/73.jpg)
Kazhdan-Lusztig-Vogan Polynomials
5 Kazhdan-Lusztig-Vogan Polynomials
5.1 Composition series: I (Γ) =∑
mΞ,ΓJ(Ξ)5.2 Character formula: J(Γ) =
∑MΞ,ΓI (Ξ)
5.3 PΞ,Γ ∈ Z[q]5.4 QΞ,Γ ∈ Z[q]5.5 mΞ,Γ = QΞ,Γ(q = 1)5.6 MΞ,Γ = ±PΞ,Γ(q = 1)
5.7 Global character (function on regular semisimple set)5.8 (Twisted KLV polynomials)
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Kazhdan-Lusztig-Vogan Polynomials
5 Kazhdan-Lusztig-Vogan Polynomials
5.1 Composition series: I (Γ) =∑
mΞ,ΓJ(Ξ)5.2 Character formula: J(Γ) =
∑MΞ,ΓI (Ξ)
5.3 PΞ,Γ ∈ Z[q]5.4 QΞ,Γ ∈ Z[q]5.5 mΞ,Γ = QΞ,Γ(q = 1)5.6 MΞ,Γ = ±PΞ,Γ(q = 1)5.7 Global character (function on regular semisimple set)
5.8 (Twisted KLV polynomials)
![Page 75: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/75.jpg)
Kazhdan-Lusztig-Vogan Polynomials
5 Kazhdan-Lusztig-Vogan Polynomials
5.1 Composition series: I (Γ) =∑
mΞ,ΓJ(Ξ)5.2 Character formula: J(Γ) =
∑MΞ,ΓI (Ξ)
5.3 PΞ,Γ ∈ Z[q]5.4 QΞ,Γ ∈ Z[q]5.5 mΞ,Γ = QΞ,Γ(q = 1)5.6 MΞ,Γ = ±PΞ,Γ(q = 1)5.7 Global character (function on regular semisimple set)5.8 (Twisted KLV polynomials)
![Page 76: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/76.jpg)
Unitary representations
6 Hermitian forms and unitary representations
6.1 Hermitian representations (preserving a Hermitian form)6.2 Unitary representations (preserving a positive definite Hermitian
form)
6.3 Gd ⊂ Gtemp ⊂ Gherm ⊂ Gunitary ⊂ Gadm
(discrete series/tempered/Hermitian/unitary/admissible)6.4 Γ ∈ Mγ → I (Γ) (standard module) J(Γ) (unique irreducible
quotient of I (Γ)
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Unitary representations
6 Hermitian forms and unitary representations
6.1 Hermitian representations (preserving a Hermitian form)6.2 Unitary representations (preserving a positive definite Hermitian
form)
6.3 Gd ⊂ Gtemp ⊂ Gherm ⊂ Gunitary ⊂ Gadm
(discrete series/tempered/Hermitian/unitary/admissible)6.4 Γ ∈ Mγ → I (Γ) (standard module) J(Γ) (unique irreducible
quotient of I (Γ)
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Unitary representations
6 Hermitian forms and unitary representations
6.1 Hermitian representations (preserving a Hermitian form)
6.2 Unitary representations (preserving a positive definite Hermitianform)
6.3 Gd ⊂ Gtemp ⊂ Gherm ⊂ Gunitary ⊂ Gadm
(discrete series/tempered/Hermitian/unitary/admissible)6.4 Γ ∈ Mγ → I (Γ) (standard module) J(Γ) (unique irreducible
quotient of I (Γ)
![Page 79: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/79.jpg)
Unitary representations
6 Hermitian forms and unitary representations
6.1 Hermitian representations (preserving a Hermitian form)6.2 Unitary representations (preserving a positive definite Hermitian
form)
6.3 Gd ⊂ Gtemp ⊂ Gherm ⊂ Gunitary ⊂ Gadm
(discrete series/tempered/Hermitian/unitary/admissible)6.4 Γ ∈ Mγ → I (Γ) (standard module) J(Γ) (unique irreducible
quotient of I (Γ)
![Page 80: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/80.jpg)
Unitary representations
6 Hermitian forms and unitary representations
6.1 Hermitian representations (preserving a Hermitian form)6.2 Unitary representations (preserving a positive definite Hermitian
form)
6.3 Gd ⊂ Gtemp ⊂ Gherm ⊂ Gunitary ⊂ Gadm
(discrete series/tempered/Hermitian/unitary/admissible)
6.4 Γ ∈ Mγ → I (Γ) (standard module) J(Γ) (unique irreducible
quotient of I (Γ)
![Page 81: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/81.jpg)
Unitary representations
6 Hermitian forms and unitary representations
6.1 Hermitian representations (preserving a Hermitian form)6.2 Unitary representations (preserving a positive definite Hermitian
form)
6.3 Gd ⊂ Gtemp ⊂ Gherm ⊂ Gunitary ⊂ Gadm
(discrete series/tempered/Hermitian/unitary/admissible)6.4 Γ ∈ Mγ → I (Γ) (standard module)
J(Γ) (unique irreducible
quotient of I (Γ)
![Page 82: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/82.jpg)
Unitary representations
6 Hermitian forms and unitary representations
6.1 Hermitian representations (preserving a Hermitian form)6.2 Unitary representations (preserving a positive definite Hermitian
form)
6.3 Gd ⊂ Gtemp ⊂ Gherm ⊂ Gunitary ⊂ Gadm
(discrete series/tempered/Hermitian/unitary/admissible)6.4 Γ ∈ Mγ → I (Γ) (standard module) J(Γ) (unique irreducible
quotient of I (Γ)
![Page 83: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/83.jpg)
Branching to K
7 Branching to K
7.1 Parametrization of K (K (R),K (C) are possibly disconnected)7.2 Compute standard module I (Γ)|K =
∑ni=1 aiµi (up to some
height bound)7.3 Compute irreducible module J(Γ)|K =
∑ni=1 aiµi (up to some
height bound)
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Branching to K
7 Branching to K
7.1 Parametrization of K (K (R),K (C) are possibly disconnected)7.2 Compute standard module I (Γ)|K =
∑ni=1 aiµi (up to some
height bound)7.3 Compute irreducible module J(Γ)|K =
∑ni=1 aiµi (up to some
height bound)
![Page 85: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/85.jpg)
Branching to K
7 Branching to K
7.1 Parametrization of K (K (R),K (C) are possibly disconnected)
7.2 Compute standard module I (Γ)|K =∑n
i=1 aiµi (up to someheight bound)
7.3 Compute irreducible module J(Γ)|K =∑n
i=1 aiµi (up to someheight bound)
![Page 86: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/86.jpg)
Branching to K
7 Branching to K
7.1 Parametrization of K (K (R),K (C) are possibly disconnected)7.2 Compute standard module I (Γ)|K =
∑ni=1 aiµi (up to some
height bound)
7.3 Compute irreducible module J(Γ)|K =∑n
i=1 aiµi (up to someheight bound)
![Page 87: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/87.jpg)
Branching to K
7 Branching to K
7.1 Parametrization of K (K (R),K (C) are possibly disconnected)7.2 Compute standard module I (Γ)|K =
∑ni=1 aiµi (up to some
height bound)7.3 Compute irreducible module J(Γ)|K =
∑ni=1 aiµi (up to some
height bound)
![Page 88: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/88.jpg)
Parabolic Induction
8 Real parabolic induction
8.1 real parabolic subgroups8.2 IndG
P (πM) =∑
aipii (in the Grothendieck group)8.3 Reducibilty of IndG
P (πM , ν) (in terms of ν)
9 Cohomological induction
9.1 theta-stable parabolic subgroups9.2 RG
q (πL) =∑
aipii (in the Grothendieck group)
9.3 More precisely:∑
i (−1)iRG ,iq (πL)
9.4 Using: coherent continuation
![Page 89: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/89.jpg)
Parabolic Induction
8 Real parabolic induction
8.1 real parabolic subgroups8.2 IndG
P (πM) =∑
aipii (in the Grothendieck group)8.3 Reducibilty of IndG
P (πM , ν) (in terms of ν)
9 Cohomological induction
9.1 theta-stable parabolic subgroups9.2 RG
q (πL) =∑
aipii (in the Grothendieck group)
9.3 More precisely:∑
i (−1)iRG ,iq (πL)
9.4 Using: coherent continuation
![Page 90: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/90.jpg)
Parabolic Induction
8 Real parabolic induction
8.1 real parabolic subgroups
8.2 IndGP (πM) =
∑aipii (in the Grothendieck group)
8.3 Reducibilty of IndGP (πM , ν) (in terms of ν)
9 Cohomological induction
9.1 theta-stable parabolic subgroups9.2 RG
q (πL) =∑
aipii (in the Grothendieck group)
9.3 More precisely:∑
i (−1)iRG ,iq (πL)
9.4 Using: coherent continuation
![Page 91: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/91.jpg)
Parabolic Induction
8 Real parabolic induction
8.1 real parabolic subgroups8.2 IndG
P (πM) =∑
aipii (in the Grothendieck group)
8.3 Reducibilty of IndGP (πM , ν) (in terms of ν)
9 Cohomological induction
9.1 theta-stable parabolic subgroups9.2 RG
q (πL) =∑
aipii (in the Grothendieck group)
9.3 More precisely:∑
i (−1)iRG ,iq (πL)
9.4 Using: coherent continuation
![Page 92: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/92.jpg)
Parabolic Induction
8 Real parabolic induction
8.1 real parabolic subgroups8.2 IndG
P (πM) =∑
aipii (in the Grothendieck group)8.3 Reducibilty of IndG
P (πM , ν) (in terms of ν)
9 Cohomological induction
9.1 theta-stable parabolic subgroups9.2 RG
q (πL) =∑
aipii (in the Grothendieck group)
9.3 More precisely:∑
i (−1)iRG ,iq (πL)
9.4 Using: coherent continuation
![Page 93: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/93.jpg)
Parabolic Induction
8 Real parabolic induction
8.1 real parabolic subgroups8.2 IndG
P (πM) =∑
aipii (in the Grothendieck group)8.3 Reducibilty of IndG
P (πM , ν) (in terms of ν)
9 Cohomological induction
9.1 theta-stable parabolic subgroups9.2 RG
q (πL) =∑
aipii (in the Grothendieck group)
9.3 More precisely:∑
i (−1)iRG ,iq (πL)
9.4 Using: coherent continuation
![Page 94: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/94.jpg)
Parabolic Induction
8 Real parabolic induction
8.1 real parabolic subgroups8.2 IndG
P (πM) =∑
aipii (in the Grothendieck group)8.3 Reducibilty of IndG
P (πM , ν) (in terms of ν)
9 Cohomological induction
9.1 theta-stable parabolic subgroups9.2 RG
q (πL) =∑
aipii (in the Grothendieck group)
9.3 More precisely:∑
i (−1)iRG ,iq (πL)
9.4 Using: coherent continuation
![Page 95: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/95.jpg)
Parabolic Induction
8 Real parabolic induction
8.1 real parabolic subgroups8.2 IndG
P (πM) =∑
aipii (in the Grothendieck group)8.3 Reducibilty of IndG
P (πM , ν) (in terms of ν)
9 Cohomological induction
9.1 theta-stable parabolic subgroups
9.2 RGq (πL) =
∑aipii (in the Grothendieck group)
9.3 More precisely:∑
i (−1)iRG ,iq (πL)
9.4 Using: coherent continuation
![Page 96: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/96.jpg)
Parabolic Induction
8 Real parabolic induction
8.1 real parabolic subgroups8.2 IndG
P (πM) =∑
aipii (in the Grothendieck group)8.3 Reducibilty of IndG
P (πM , ν) (in terms of ν)
9 Cohomological induction
9.1 theta-stable parabolic subgroups9.2 RG
q (πL) =∑
aipii (in the Grothendieck group)
9.3 More precisely:∑
i (−1)iRG ,iq (πL)
9.4 Using: coherent continuation
![Page 97: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/97.jpg)
Parabolic Induction
8 Real parabolic induction
8.1 real parabolic subgroups8.2 IndG
P (πM) =∑
aipii (in the Grothendieck group)8.3 Reducibilty of IndG
P (πM , ν) (in terms of ν)
9 Cohomological induction
9.1 theta-stable parabolic subgroups9.2 RG
q (πL) =∑
aipii (in the Grothendieck group)
9.3 More precisely:
∑i (−1)iRG ,i
q (πL)9.4 Using: coherent continuation
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Parabolic Induction
8 Real parabolic induction
8.1 real parabolic subgroups8.2 IndG
P (πM) =∑
aipii (in the Grothendieck group)8.3 Reducibilty of IndG
P (πM , ν) (in terms of ν)
9 Cohomological induction
9.1 theta-stable parabolic subgroups9.2 RG
q (πL) =∑
aipii (in the Grothendieck group)
9.3 More precisely:∑
i (−1)iRG ,iq (πL)
9.4 Using: coherent continuation
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Parabolic Induction
8 Real parabolic induction
8.1 real parabolic subgroups8.2 IndG
P (πM) =∑
aipii (in the Grothendieck group)8.3 Reducibilty of IndG
P (πM , ν) (in terms of ν)
9 Cohomological induction
9.1 theta-stable parabolic subgroups9.2 RG
q (πL) =∑
aipii (in the Grothendieck group)
9.3 More precisely:∑
i (−1)iRG ,iq (πL)
9.4 Using: coherent continuation
![Page 100: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/100.jpg)
Parabolic Induction
8 Real parabolic induction
8.1 real parabolic subgroups8.2 IndG
P (πM) =∑
aipii (in the Grothendieck group)8.3 Reducibilty of IndG
P (πM , ν) (in terms of ν)
9 Cohomological induction
9.1 theta-stable parabolic subgroups9.2 RG
q (πL) =∑
aipii (in the Grothendieck group)
9.3 More precisely:∑
i (−1)iRG ,iq (πL)
9.4 Using: coherent continuation
![Page 101: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/101.jpg)
Example G = Sp(12,R), M = GL(5,R)× SL(2,R)
atlas> G:=Sp(12,R)
Value: connected split real group with Lie algebra ’sp(12,R)’
atlas> set real_parabolics=all_real_parabolics (G)
Variable real_parabolics: [KGPElt]
atlas> #real_parabolics
Value: 64
atlas> void:for P@i in real_parabolics do if
ss_rank (Levi(P))=5 then prints(i, " ", Levi(P)) fi od
31 sl(6,R).gl(1,R)
47 sl(5,R).sl(2,R).gl(1,R)
55 sl(4,R).sp(4,R).gl(1,R)
59 sl(3,R).sp(6,R).gl(1,R)
61 sl(2,R).sp(8,R).gl(1,R)
62 sp(10,R).gl(1,R)
atlas> set P=real_parabolics[47]
Variable P: KGPElt
atlas> real_induce_irreducible(trivial(Levi(P)),G)
Value:
1*parameter(x=4898,lambda=[6,5,4,3,2,1]/1,nu=[2,2,1,1,1,0]/1)
1*parameter(x=4117,lambda=[5,6,1,3,4,0]/1,nu=[3,4,0,2,3,0]/2)
1*parameter(x=4116,lambda=[5,6,1,3,4,0]/1,nu=[3,4,0,2,3,0]/2)
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Nilpotent Orbits
10. Nilpotent orbits
10.1 parametrize Complex nilpotent orbits10.2 parametrize real nilpotent orbits10.3 Note: These parametrizations are case-free, in atlas language10.4 complex nilpotent: identity component of reductive part of
centralizer, component group10.5 coming soon: real nilpotent: identity component of reductive
part of centralizer, component group
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Nilpotent Orbits
10. Nilpotent orbits
10.1 parametrize Complex nilpotent orbits10.2 parametrize real nilpotent orbits10.3 Note: These parametrizations are case-free, in atlas language10.4 complex nilpotent: identity component of reductive part of
centralizer, component group10.5 coming soon: real nilpotent: identity component of reductive
part of centralizer, component group
![Page 104: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/104.jpg)
Nilpotent Orbits
10. Nilpotent orbits
10.1 parametrize Complex nilpotent orbits
10.2 parametrize real nilpotent orbits10.3 Note: These parametrizations are case-free, in atlas language10.4 complex nilpotent: identity component of reductive part of
centralizer, component group10.5 coming soon: real nilpotent: identity component of reductive
part of centralizer, component group
![Page 105: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/105.jpg)
Nilpotent Orbits
10. Nilpotent orbits
10.1 parametrize Complex nilpotent orbits10.2 parametrize real nilpotent orbits
10.3 Note: These parametrizations are case-free, in atlas language10.4 complex nilpotent: identity component of reductive part of
centralizer, component group10.5 coming soon: real nilpotent: identity component of reductive
part of centralizer, component group
![Page 106: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/106.jpg)
Nilpotent Orbits
10. Nilpotent orbits
10.1 parametrize Complex nilpotent orbits10.2 parametrize real nilpotent orbits10.3 Note: These parametrizations are case-free, in atlas language
10.4 complex nilpotent: identity component of reductive part ofcentralizer, component group
10.5 coming soon: real nilpotent: identity component of reductivepart of centralizer, component group
![Page 107: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/107.jpg)
Nilpotent Orbits
10. Nilpotent orbits
10.1 parametrize Complex nilpotent orbits10.2 parametrize real nilpotent orbits10.3 Note: These parametrizations are case-free, in atlas language10.4 complex nilpotent: identity component of reductive part of
centralizer, component group
10.5 coming soon: real nilpotent: identity component of reductivepart of centralizer, component group
![Page 108: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/108.jpg)
Nilpotent Orbits
10. Nilpotent orbits
10.1 parametrize Complex nilpotent orbits10.2 parametrize real nilpotent orbits10.3 Note: These parametrizations are case-free, in atlas language10.4 complex nilpotent: identity component of reductive part of
centralizer, component group10.5 coming soon: real nilpotent: identity component of reductive
part of centralizer, component group
![Page 109: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/109.jpg)
Advanced topics
11. Advanced topics (selected)
11.1 Weyl character formula11.2 Frobenius-Schur and quaternionic indicators for finite
dimensional representation11.3 Conjugacy classes in W (nice representatives)11.4 Character table of W11.5 Coherent continuation representation of W11.6 Cell representations11.7 Special representations of W
![Page 110: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/110.jpg)
Advanced topics
11. Advanced topics (selected)
11.1 Weyl character formula11.2 Frobenius-Schur and quaternionic indicators for finite
dimensional representation11.3 Conjugacy classes in W (nice representatives)11.4 Character table of W11.5 Coherent continuation representation of W11.6 Cell representations11.7 Special representations of W
![Page 111: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/111.jpg)
Advanced topics
11. Advanced topics (selected)
11.1 Weyl character formula
11.2 Frobenius-Schur and quaternionic indicators for finitedimensional representation
11.3 Conjugacy classes in W (nice representatives)11.4 Character table of W11.5 Coherent continuation representation of W11.6 Cell representations11.7 Special representations of W
![Page 112: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/112.jpg)
Advanced topics
11. Advanced topics (selected)
11.1 Weyl character formula11.2 Frobenius-Schur and quaternionic indicators for finite
dimensional representation
11.3 Conjugacy classes in W (nice representatives)11.4 Character table of W11.5 Coherent continuation representation of W11.6 Cell representations11.7 Special representations of W
![Page 113: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/113.jpg)
Advanced topics
11. Advanced topics (selected)
11.1 Weyl character formula11.2 Frobenius-Schur and quaternionic indicators for finite
dimensional representation11.3 Conjugacy classes in W (nice representatives)
11.4 Character table of W11.5 Coherent continuation representation of W11.6 Cell representations11.7 Special representations of W
![Page 114: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/114.jpg)
Advanced topics
11. Advanced topics (selected)
11.1 Weyl character formula11.2 Frobenius-Schur and quaternionic indicators for finite
dimensional representation11.3 Conjugacy classes in W (nice representatives)11.4 Character table of W
11.5 Coherent continuation representation of W11.6 Cell representations11.7 Special representations of W
![Page 115: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/115.jpg)
Advanced topics
11. Advanced topics (selected)
11.1 Weyl character formula11.2 Frobenius-Schur and quaternionic indicators for finite
dimensional representation11.3 Conjugacy classes in W (nice representatives)11.4 Character table of W11.5 Coherent continuation representation of W
11.6 Cell representations11.7 Special representations of W
![Page 116: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/116.jpg)
Advanced topics
11. Advanced topics (selected)
11.1 Weyl character formula11.2 Frobenius-Schur and quaternionic indicators for finite
dimensional representation11.3 Conjugacy classes in W (nice representatives)11.4 Character table of W11.5 Coherent continuation representation of W11.6 Cell representations
11.7 Special representations of W
![Page 117: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/117.jpg)
Advanced topics
11. Advanced topics (selected)
11.1 Weyl character formula11.2 Frobenius-Schur and quaternionic indicators for finite
dimensional representation11.3 Conjugacy classes in W (nice representatives)11.4 Character table of W11.5 Coherent continuation representation of W11.6 Cell representations11.7 Special representations of W
![Page 118: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/118.jpg)
11. Advanced topics (continued)
11.7 AVann(π) “complex associated variety”, closure of a singlecomplex orbit (π irreducible)
11.8 Primitive ideals11.9 AV (π) “real associated variety”: {O1, . . . ,Oi} nilpotent orbits
of G (R) on gR or K orbits on s11.10 “Weak” Arthur packets of special unipotent representations11.11 “Honest” Arthur packets of special unipotent representations
12. p-adic groups. . . (Sorry, just kidding)
Thank You
![Page 119: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/119.jpg)
11. Advanced topics (continued)
11.7 AVann(π) “complex associated variety”, closure of a singlecomplex orbit (π irreducible)
11.8 Primitive ideals11.9 AV (π) “real associated variety”: {O1, . . . ,Oi} nilpotent orbits
of G (R) on gR or K orbits on s11.10 “Weak” Arthur packets of special unipotent representations11.11 “Honest” Arthur packets of special unipotent representations
12. p-adic groups. . . (Sorry, just kidding)
Thank You
![Page 120: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/120.jpg)
11. Advanced topics (continued)
11.7 AVann(π) “complex associated variety”, closure of a singlecomplex orbit (π irreducible)
11.8 Primitive ideals11.9 AV (π) “real associated variety”: {O1, . . . ,Oi} nilpotent orbits
of G (R) on gR or K orbits on s11.10 “Weak” Arthur packets of special unipotent representations11.11 “Honest” Arthur packets of special unipotent representations
12. p-adic groups. . . (Sorry, just kidding)
Thank You
![Page 121: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/121.jpg)
11. Advanced topics (continued)
11.7 AVann(π) “complex associated variety”, closure of a singlecomplex orbit (π irreducible)
11.8 Primitive ideals
11.9 AV (π) “real associated variety”: {O1, . . . ,Oi} nilpotent orbitsof G (R) on gR or K orbits on s
11.10 “Weak” Arthur packets of special unipotent representations11.11 “Honest” Arthur packets of special unipotent representations
12. p-adic groups. . . (Sorry, just kidding)
Thank You
![Page 122: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/122.jpg)
11. Advanced topics (continued)
11.7 AVann(π) “complex associated variety”, closure of a singlecomplex orbit (π irreducible)
11.8 Primitive ideals11.9 AV (π) “real associated variety”: {O1, . . . ,Oi} nilpotent orbits
of G (R) on gR or K orbits on s
11.10 “Weak” Arthur packets of special unipotent representations11.11 “Honest” Arthur packets of special unipotent representations
12. p-adic groups. . . (Sorry, just kidding)
Thank You
![Page 123: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/123.jpg)
11. Advanced topics (continued)
11.7 AVann(π) “complex associated variety”, closure of a singlecomplex orbit (π irreducible)
11.8 Primitive ideals11.9 AV (π) “real associated variety”: {O1, . . . ,Oi} nilpotent orbits
of G (R) on gR or K orbits on s11.10 “Weak” Arthur packets of special unipotent representations
11.11 “Honest” Arthur packets of special unipotent representations
12. p-adic groups. . . (Sorry, just kidding)
Thank You
![Page 124: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/124.jpg)
11. Advanced topics (continued)
11.7 AVann(π) “complex associated variety”, closure of a singlecomplex orbit (π irreducible)
11.8 Primitive ideals11.9 AV (π) “real associated variety”: {O1, . . . ,Oi} nilpotent orbits
of G (R) on gR or K orbits on s11.10 “Weak” Arthur packets of special unipotent representations11.11 “Honest” Arthur packets of special unipotent representations
12. p-adic groups. . . (Sorry, just kidding)
Thank You
![Page 125: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/125.jpg)
11. Advanced topics (continued)
11.7 AVann(π) “complex associated variety”, closure of a singlecomplex orbit (π irreducible)
11.8 Primitive ideals11.9 AV (π) “real associated variety”: {O1, . . . ,Oi} nilpotent orbits
of G (R) on gR or K orbits on s11.10 “Weak” Arthur packets of special unipotent representations11.11 “Honest” Arthur packets of special unipotent representations
12. p-adic groups. . .
(Sorry, just kidding)
Thank You
![Page 126: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/126.jpg)
11. Advanced topics (continued)
11.7 AVann(π) “complex associated variety”, closure of a singlecomplex orbit (π irreducible)
11.8 Primitive ideals11.9 AV (π) “real associated variety”: {O1, . . . ,Oi} nilpotent orbits
of G (R) on gR or K orbits on s11.10 “Weak” Arthur packets of special unipotent representations11.11 “Honest” Arthur packets of special unipotent representations
12. p-adic groups. . . (Sorry, just kidding)
Thank You
![Page 127: Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and Representations Je rey Adams Sun Yat-sen University, Guanhzhou August 4, 2018 Slides available](https://reader034.vdocument.in/reader034/viewer/2022051811/601c6d592330df4d790e7b8d/html5/thumbnails/127.jpg)
11. Advanced topics (continued)
11.7 AVann(π) “complex associated variety”, closure of a singlecomplex orbit (π irreducible)
11.8 Primitive ideals11.9 AV (π) “real associated variety”: {O1, . . . ,Oi} nilpotent orbits
of G (R) on gR or K orbits on s11.10 “Weak” Arthur packets of special unipotent representations11.11 “Honest” Arthur packets of special unipotent representations
12. p-adic groups. . . (Sorry, just kidding)
Thank You