lecture 6: cells and orbits -...

Lecture 6: Cells and Orbits

Birne Binegar

Department of MathematicsOklahoma State UniversityStillwater, OK 74078, USA

Nankai Summer School in Representation Theory and Harmonic AnalysisJune, 2008

Birne Binegar (Oklahoma State University) Lecture 6: Cells and Orbits Nankai 2008 1 / 24

The Atlas Setting

GR : a real reductive Lie group realizable as the set of real points of a reductivealgebraic group defined over R;

GR,adm : set of equivalence classes of irr admissible reps

Lλ : a set of Langlands parameters for irr admissible reps of regular integralinfinitesimal character λ (a finite set).

HCλ ={Vx = πx |K -finite | x ∈ Lλ

}: set of irreducible Harish-Chandra modules

corresponding to the irr admissible reps πx ∈ GR, x ∈ Lλ.

The Atlas software catalogs and analyzes reps in HCλ.


Notation / Apparatus

g = Lie (GR)C; h, a CSA for g;∆ = ∆ (h, g), roots of h in g;Π ⊂ ∆, choice of simple roots in ∆;

G : adjoint group of g

Ng : nilpotent cone in g (identifying g∗ with g)

Set S ≡ {special nilpotent orbits}d : G\Ng → G\Ng : the Spaltenstein-Barbasch-Vogan duality map that restricts toan involution on image(d) ≡ S ≡ set of special nilpotent orbits.


Cells of Harish-Chandra modules

Definition: Let x , y ∈ HCλ. Write x ⇀ y if there exists a f.d. rep F occurring in T (g)such that

y occurs as subquotient of x ⊗ F

A cell of H-C modules is a maximal collection of x ∈ HCλ such that

x , y ∈ C =⇒ x ⇀ y and y ⇀ x

Basic facts:

(i) x ∈ HCλ ⇒ AnnU(g)(x) primitive ideal of reg int char⇒ gr(Ann(x)) ∼ ideal in S(g⇒ associated variety AV (Ann(x)) ∈ g∗

Fact: λ reg integral ⇒ AV (Ann(x)) = O the closure of a special orbit

(ii) x , y ∈ C =⇒ Ox = Oy ≡ OC

(iii) x ∈ Lλ, with λ regular, integral inf. char.=⇒ Ox is special nilpotent orbit.


Problem: which cells of reps correspond to which special nilpotent orbits?

Key Fact:

The W -rep carried by a cell (induced from coherent cont rep on block) has a uniquespecial constituent σC .

This coincides with special W -rep attached to OC via Springer correspondence.


Atlas Output

The Atlas software not only catalogs the KLV polynomials for the representations in Lρ,it computes the entire W -graph of Lρ: a weighted directed graph such that

vertices ↔ x ∈ Lρvertex weights ↔ descent sets τ(x) of x ∈ LρFor each x ∈ Lλ, τ(x) is a certain subset of Πτ(x) is the tau invariant of Ann(Vx).

edges ↔ relations y → x ≡ Vy occurs in Vx ⊗ g

edge multiplicities: mult(y → x) = multiplicity of Vy in Vx ⊗ g

H-C cells correspond to bidirectionally connected subgraphs


Example: the big block of the split real form of G2.

block element descent set (edge vertex,multiplicity)0 {} {}1 {2} {(3,1)}2 {1} {(4,1)}3 {1} {(0,1),(1,1),(6,1)}4 {2} {(0,1),(2,1),(5,1)}5 {1} {(4,1),(8,1)}6 {2} {(3,1),(7,1)}7 {1} {(6,1),(11,1)}8 {2} {(5,1),(10,1)}9 {1,2} {(7,1),(8,1)}10 {1} {(8,1)}11 {2} {(7,1)}


The W -graph for this block thus looks like

11, {2}

1��

9, {1, 2}

1{{vvvvv 1

##HHHHH 10, {1}

1��

7, {1}

1

OO

1��

8, {2}

1��

1

OO

6, {2}

1

OO

1��

5, {1}

1��

1

OO

1, {2}1 //

3, {1}1oo

1

OO

1

##HHHHH 4, {2}

1{{vvvvv

1

OO

1 // 2, {1}1oo

0, {}

Cell # Members0 01 1, 3, 6, 7, 112 2, 4, 5, 8, 103 9


The Spaltenstein-Vogan Criterion

Theorem. (Spaltenstein, Vogan) Suppose C is a cell of H-C modules with associatedspecial nilpotent orbit OC and let l be a (standard) Levi subalgebra of g. Then

OC ⊂ indgl (0l) ⇐⇒ ∃ x ∈ C s.t. Πl ⊂ τ(x)

where Πl = the simple roots of l. Here Πl ⊂ Πg and

indgl (0l) ≡ unique dense orbit in G · n

where n is nilradical of any parabolic subalgebra of g with Levi factor l.

Orbits of the form indgl (0l) are called Richardson orbits.

Upshot: tau invariants of a cell constrain which Richardson orbit closures can contain OC


Problem: Every Richardson orbit is special, but not every special orbit is Richardson.

How do we separate configurations like

OC = ORichardson

OC ′ = Ospecial

OO

S-V criterion would only tell us that both OC and OC ′ are contained in ORichardson


Levi subalgebras and Richardson orbits

Γ ⊂ Π : a subset of the simple roots.

lΓ : standard Levi subalgebra attached to

lΓ = h +∑α∈〈Γ〉

gα

RΓ = indglΓ

(0lΓ ) : the Richardson orbit induced from the trivial orbit of a Levisubalgebra lΓ of g

Fact: every special orbit O is determined by

(i) the Richardson orbits that contain O(ii) the Richardon orbits that contain d(O)

David Vogan’s Idea: The tau invariants of a cell should tell us which Richardson orbitscontain OC and which Richardson orbits contain the SBV dual of OC .


Tau signatures for cells

Setτ(C) ≡ {τ(x) | x ∈ C}

Facts

# distinct τ(C) = # special nilpotent orbits

Letτ∨(C) = {Π− τ(x) | x ∈ C}

=⇒ duality operation for tau sets.


Definition:

Ψ = {Γ ⊂ Π} : a set of standard Γ’s: a collection of Γ ∈ 2Π such that

i : Ψ↔ {conjugacy classes of Levi subalebras}

is a bijection. (E.g., choose std Γ’s to be first in the lexicographic ordering of theirW -conj class)

Let Γ, Γ′ ∈ Ψ and let lΓ and lΓ′ be the corresponding standard Levi subalgebras of g. Weshall say

Γ ≤ Γ′ ⇐⇒ indglΓ

(0) ⊂ indglΓ′

(0)

Remark: this ordering tends to reverse the ordering by cardinality.

Definition: The tau signature of an H-C cell C is the pair

τsig (C) ≡(min (τ(C) ∩Ψ) , min

(τ∨(C) ∩Ψ

))


Tau signatures for Special Orbits

Definition: Let O be a special orbit. The tau signature of O is the pair (τ (O) , τ∨ (O))where

τ (O) = min{

Γ ∈ Ψ | O ⊂ indglΓ

(0lΓ )}

τ∨ (O) = min{

Γ ∈ Ψ | d (O) ⊂ indglΓ

(0lΓ )}

The point: we are using pairs of subsets of simple roots to tell us when a Richardsonorbit closure can contain a special orbit (or its dual).

The same kind of pairs tells us when the orbit attached to a cell can be contained inRichardson orbit (or when the dual cell can be contained in the closure of Richardsonorbit).

Corollary (to S-V criterion)

OC = O ⇐⇒ τsig (C) = τsig (O)


Example: Special Orbits of D5

O[9,1] hh

vv

O[7,3]

rrr KKK44

**

O[7,13]

KKK::

$$

O[52]

ttt dd

zz

O[5,3,12]

sss

8888888 ii

uu

O[42,12]

99

%%

O[33,1]

O[5,15]

��O

[32,22]

KKKO

[32,14]

sss JJJO

[24,12]

KKKO

[3,17]

tttO

[22,16]

O[110]


Richardson Orbits of D5

O[9,1] = R{}hh

vv

O[7,3] = R{1}

hhhhhhhh VVVVV44

**

O[7,13]

VVVVVVV::

$$

O[52]

= R{1,3}

hhhhh dd

zz

O[5,3,12]

= R{1,4,5}

hhhhh

MMMMMMMMMMM ii

uu

O[42,12]

= R{1,2,4}99

%%

O[33,1]

= R{1,2,4,5} O[5,15]

qqqqqqqqqqqO

[32,22]

VVVVVVV

O[32,14]

= R{1,3,4,5}

hhhhh VVVVVO

[24,12]= R{1,2,3,4}

VVVVVVVO

[3,17]= R{2,3,4,5}

hhhhhhhO

[22,16]

O[110]

= R{1,2,3,4,5}


Tau Signatures of Special Orbits of D5

O{};{1,2,3,4,5}hh

vv

O{1};{1,2,3,4},{2,3,4,5}

ggggg VVVVV44

**

O{1};{2,3,4,5}

WWWWW::

$$

O{1,3};{1,2,3,4}

hhhhh dd

zz

O{1,4,5};{1,3,4,5}

ggggg

MMMMMMMMMMM ii

uu

O{1,2,4};{1,2,3}88

&&

O{1,2,4,5};{1,2,4,5} O{1,4,5};{1,4,5}

qqqqqqqqqqqO{1,3,4,5};{1,4,5}

WWWWWO{1,3,4,5};{1,4,5}

ggggg VVVVVO{1,2,3,4};{1,3}

WWWWWO{2,3,4,5};{1}

hhhhhO{1,2,3,4},{2,3,4,5};{1}

O{1,2,3,4,5};{}


Tau signatures for cells in the big block of SO(5, 5)• 365 representations with inf. char. ρ in big block• 32 cells in the big block

Output of extract-cells

// Individual cells.

// cell #0:

0[0]: {}

// cell #1:

0[1]: {2} --> 1,2

1[3]: {1} --> 0

2[5]: {3} --> 0,3,4

3[13]: {5} --> 2

4[14]: {4} --> 2

*

*

*

// cell #29:

0[328]: {1,2,4,5} --> 2,3

1[340]: {2,3,4,5} --> 2

2[358]: {1,3,4,5} --> 0,1

3[364]: {1,2,3} --> 0

// cell #30:

0[353]: {1,2,3,4,5}

// cell #31:

0[357]: {1,2,3,4,5}


cell # tau signature

0 {} , {1,2,3,4,5}

1 {1} , {1,2,3,4}

2 {1} , {2,3,4,5}

3 {1,3} , {1,3,4,5}

* *

* *

* *

28 {2,3,4,5} , {1}

29 {2,3,4,5} , {1}

30 {1,2,3,4,5} , {}

31 {1,2,3,4,5} , {}

Each of these coincides with the tau signature of a particular nilpotent orbit.


Cell-Orbit Correspondences for SO(5, 5)

{0}

{1}

llllllRRRRRR

{2, 5}

RRRRR {3, 4}

lllll{6, 7, 15, 17}

lllll

DDDDDDDD

{8, 9}

{10, 12, 13} {11, 21}

zzzzzzzz{16, 18}

RRRR

{14, 19, 20, 22}

llll RRRR

{24, 25}

RRRR {23, 28, 29}

llll{26, 27}

{30, 31}


More Generally:

Exceptional Groups: tables by Spaltenstein list induced orbits, and Hasse diagrams.

Tau signatures of special orbits can be done by hand.

1. Use Spaltenstein’s tables to figure out which special orbits are Richardson orbits andto identify the std Γ’s corresponding to the corresponding Levi subalgebra.

2. Place the Richardson orbits in the Hasse diagram of special orbits, and then figure outthe Γ parameters of the minimal Richardson orbits that contain a given special orbit andthe minimal Richardson orbits that contain its Spaltenstein dual

Even E8 can be done by hand.

Classical Groups:

Partition classification −→ closure relations

Just need to

which partitions correspond to special orbits (easy recipes in Collingwood-McGovern)use dominance ordering of partitions to partial order special orbitsuse formulas in [C-M] to determine partitions corresponding to Richardson orbits foreach Γ ∈ Ψ. Place these in the Hasse diagram of special orbits and at the same timepartial order Ψ.Use the partial ordering of Ψ to ascribe tau signatures to cells (employing atlas

data)match orbit tau sigs to cell tau sigs

Cell-Orbit correspondences have now been computed for all exceptional and classicalcases up to rank 8.


Conclusion:

Atlas data =⇒ algorithm mapping Langlands parameters to nilpotent orbits.Key is to first collect Langlands parameters into cells.

Can one actually identify even finer invariants?Can one tell when Ann(Vx ) = Ann(Vy )? (yes!).What about the associated variety of Vx (union of KC-orbits)?

Are there representation theoretical intepretations of other combinatorial aspects ofW -graphs?


Some References

D. Barbasch and D. Vogan, Unipotent representations of complex semisimple groups, Ann.of Math 121 (1985), 41-110.

D. Collingwood and W. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, VanNostrand Reinhold, New York, 1993.

N. Spaltenstein, Classes unipotentes et sous-groups de Borel, Lec. Notes in Math. 946,Springer-Verlag, New York, 1982.

N. Spaltenstein, A property of special representations of Weyl groups, J. Reine Angew.Math. 343 (1983), 212-220.

D. Vogan, Representations of Real Reductive Lie Groups, Birkhauser, Boston, 1981


Acknowledgements

Supported by the National Science Foundation for support via the Atlas for Lie Groupsand Representations FRG (DMS 0554278).


lecture 6: cells and orbits -...

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