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Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha Sahi Conference on L-functions, Jeju August 2012 Gourevitch-Sahi Degenerate Whittaker functionals August 2012 1 / 14

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Page 1: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Degenerate Whittaker functionals for real reductivegroups

Dmitry Gourevitch & Siddhartha SahiConference on L-functions, Jeju

August 2012

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 1 / 14

Page 2: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Whittaker functionals

F: local field of char.0, G : reductive group over F.

M(G ): smooth admissible representations (of moderate growth).

Assume G quasisplit: fix Borel B = HN, n := LieF(N).

Define n′ = [n, n] , v = n/n′, Ψ = v∗ ⊂ n∗

Ψ ←→ Lie algebra characters of n ←→ unitary group characters of N

Ψ ⊃ Ψ× := non-degenerate characters.

For ψ ∈ Ψ, π ∈ M(G ) define

Wh∗ψ(π) := HomctsN (π, ψ), Ψ(π) := {ψ ∈ Ψ : Wh∗ψ(π) 6= 0}

(Casselman) For any ψ ∈ Ψ×, π 7→ Wh∗ψ(π) is an exact functor.

Theorem (Gelfand-Kazhdan, Shalika)

For π ∈ Irr(G ), ψ ∈ Ψ×, dim Wh∗ψ(π) ≤ 1.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 2 / 14

Page 3: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Whittaker functionals

F: local field of char.0, G : reductive group over F.

M(G ): smooth admissible representations (of moderate growth).

Assume G quasisplit: fix Borel B = HN, n := LieF(N).

Define n′ = [n, n] , v = n/n′, Ψ = v∗ ⊂ n∗

Ψ ←→ Lie algebra characters of n ←→ unitary group characters of N

Ψ ⊃ Ψ× := non-degenerate characters.

For ψ ∈ Ψ, π ∈ M(G ) define

Wh∗ψ(π) := HomctsN (π, ψ), Ψ(π) := {ψ ∈ Ψ : Wh∗ψ(π) 6= 0}

(Casselman) For any ψ ∈ Ψ×, π 7→ Wh∗ψ(π) is an exact functor.

Theorem (Gelfand-Kazhdan, Shalika)

For π ∈ Irr(G ), ψ ∈ Ψ×, dim Wh∗ψ(π) ≤ 1.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 2 / 14

Page 4: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Whittaker functionals

F: local field of char.0, G : reductive group over F.

M(G ): smooth admissible representations (of moderate growth).

Assume G quasisplit: fix Borel B = HN, n := LieF(N).

Define n′ = [n, n] , v = n/n′, Ψ = v∗ ⊂ n∗

Ψ ←→ Lie algebra characters of n ←→ unitary group characters of N

Ψ ⊃ Ψ× := non-degenerate characters.

For ψ ∈ Ψ, π ∈ M(G ) define

Wh∗ψ(π) := HomctsN (π, ψ), Ψ(π) := {ψ ∈ Ψ : Wh∗ψ(π) 6= 0}

(Casselman) For any ψ ∈ Ψ×, π 7→ Wh∗ψ(π) is an exact functor.

Theorem (Gelfand-Kazhdan, Shalika)

For π ∈ Irr(G ), ψ ∈ Ψ×, dim Wh∗ψ(π) ≤ 1.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 2 / 14

Page 5: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Whittaker functionals

F: local field of char.0, G : reductive group over F.

M(G ): smooth admissible representations (of moderate growth).

Assume G quasisplit: fix Borel B = HN, n := LieF(N).

Define n′ = [n, n] , v = n/n′, Ψ = v∗ ⊂ n∗

Ψ ←→ Lie algebra characters of n ←→ unitary group characters of N

Ψ ⊃ Ψ× := non-degenerate characters.

For ψ ∈ Ψ, π ∈ M(G ) define

Wh∗ψ(π) := HomctsN (π, ψ), Ψ(π) := {ψ ∈ Ψ : Wh∗ψ(π) 6= 0}

(Casselman) For any ψ ∈ Ψ×, π 7→ Wh∗ψ(π) is an exact functor.

Theorem (Gelfand-Kazhdan, Shalika)

For π ∈ Irr(G ), ψ ∈ Ψ×, dim Wh∗ψ(π) ≤ 1.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 2 / 14

Page 6: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Whittaker functionals

F: local field of char.0, G : reductive group over F.

M(G ): smooth admissible representations (of moderate growth).

Assume G quasisplit: fix Borel B = HN, n := LieF(N).

Define n′ = [n, n] , v = n/n′, Ψ = v∗ ⊂ n∗

Ψ ←→ Lie algebra characters of n ←→ unitary group characters of N

Ψ ⊃ Ψ× := non-degenerate characters.

For ψ ∈ Ψ, π ∈ M(G ) define

Wh∗ψ(π) := HomctsN (π, ψ), Ψ(π) := {ψ ∈ Ψ : Wh∗ψ(π) 6= 0}

(Casselman) For any ψ ∈ Ψ×, π 7→ Wh∗ψ(π) is an exact functor.

Theorem (Gelfand-Kazhdan, Shalika)

For π ∈ Irr(G ), ψ ∈ Ψ×, dim Wh∗ψ(π) ≤ 1.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 2 / 14

Page 7: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Whittaker functionals

F: local field of char.0, G : reductive group over F.

M(G ): smooth admissible representations (of moderate growth).

Assume G quasisplit: fix Borel B = HN, n := LieF(N).

Define n′ = [n, n] , v = n/n′, Ψ = v∗ ⊂ n∗

Ψ ←→ Lie algebra characters of n ←→ unitary group characters of N

Ψ ⊃ Ψ× := non-degenerate characters.

For ψ ∈ Ψ, π ∈ M(G ) define

Wh∗ψ(π) := HomctsN (π, ψ), Ψ(π) := {ψ ∈ Ψ : Wh∗ψ(π) 6= 0}

(Casselman) For any ψ ∈ Ψ×, π 7→ Wh∗ψ(π) is an exact functor.

Theorem (Gelfand-Kazhdan, Shalika)

For π ∈ Irr(G ), ψ ∈ Ψ×, dim Wh∗ψ(π) ≤ 1.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 2 / 14

Page 8: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Whittaker functionals

F: local field of char.0, G : reductive group over F.

M(G ): smooth admissible representations (of moderate growth).

Assume G quasisplit: fix Borel B = HN, n := LieF(N).

Define n′ = [n, n] , v = n/n′, Ψ = v∗ ⊂ n∗

Ψ ←→ Lie algebra characters of n ←→ unitary group characters of N

Ψ ⊃ Ψ× := non-degenerate characters.

For ψ ∈ Ψ, π ∈ M(G ) define

Wh∗ψ(π) := HomctsN (π, ψ), Ψ(π) := {ψ ∈ Ψ : Wh∗ψ(π) 6= 0}

(Casselman) For any ψ ∈ Ψ×, π 7→ Wh∗ψ(π) is an exact functor.

Theorem (Gelfand-Kazhdan, Shalika)

For π ∈ Irr(G ), ψ ∈ Ψ×, dim Wh∗ψ(π) ≤ 1.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 2 / 14

Page 9: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Whittaker functionals

F: local field of char.0, G : reductive group over F.

M(G ): smooth admissible representations (of moderate growth).

Assume G quasisplit: fix Borel B = HN, n := LieF(N).

Define n′ = [n, n] , v = n/n′, Ψ = v∗ ⊂ n∗

Ψ ←→ Lie algebra characters of n ←→ unitary group characters of N

Ψ ⊃ Ψ× := non-degenerate characters.

For ψ ∈ Ψ, π ∈ M(G ) define

Wh∗ψ(π) := HomctsN (π, ψ), Ψ(π) := {ψ ∈ Ψ : Wh∗ψ(π) 6= 0}

(Casselman) For any ψ ∈ Ψ×, π 7→ Wh∗ψ(π) is an exact functor.

Theorem (Gelfand-Kazhdan, Shalika)

For π ∈ Irr(G ), ψ ∈ Ψ×, dim Wh∗ψ(π) ≤ 1.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 2 / 14

Page 10: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Kostant’s theorem

We say π is generic if ∃ψ ∈ Ψ× s.t. Whψ(π) 6= 0.

Theorem (Kostant, Rodier)

π is generic iff it is large.

Theorem (Harish-Chandra, Howe)

Near e ∈ G , the character distribution (asymptotically) equals to a linearcombination of Fourier transforms of Haar measures of nilpotent coadjointorbits.

χπ ≈∑ aOF (µO)

Define WF(π) = ∪{O | aO 6= 0} ⊂ N , where N ⊂ g∗ denotes thenilpotent cone.

π is called large if WF(π) = N .

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 3 / 14

Page 11: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Kostant’s theorem

We say π is generic if ∃ψ ∈ Ψ× s.t. Whψ(π) 6= 0.

Theorem (Kostant, Rodier)

π is generic iff it is large.

Theorem (Harish-Chandra, Howe)

Near e ∈ G , the character distribution (asymptotically) equals to a linearcombination of Fourier transforms of Haar measures of nilpotent coadjointorbits.

χπ ≈∑ aOF (µO)

Define WF(π) = ∪{O | aO 6= 0} ⊂ N , where N ⊂ g∗ denotes thenilpotent cone.

π is called large if WF(π) = N .

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 3 / 14

Page 12: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Kostant’s theorem

We say π is generic if ∃ψ ∈ Ψ× s.t. Whψ(π) 6= 0.

Theorem (Kostant, Rodier)

π is generic iff it is large.

Theorem (Harish-Chandra, Howe)

Near e ∈ G , the character distribution (asymptotically) equals to a linearcombination of Fourier transforms of Haar measures of nilpotent coadjointorbits.

χπ ≈∑ aOF (µO)

Define WF(π) = ∪{O | aO 6= 0} ⊂ N , where N ⊂ g∗ denotes thenilpotent cone.

π is called large if WF(π) = N .

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 3 / 14

Page 13: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Kostant’s theorem

We say π is generic if ∃ψ ∈ Ψ× s.t. Whψ(π) 6= 0.

Theorem (Kostant, Rodier)

π is generic iff it is large.

Theorem (Harish-Chandra, Howe)

Near e ∈ G , the character distribution (asymptotically) equals to a linearcombination of Fourier transforms of Haar measures of nilpotent coadjointorbits.

χπ ≈∑ aOF (µO)

Define WF(π) = ∪{O | aO 6= 0} ⊂ N , where N ⊂ g∗ denotes thenilpotent cone.

π is called large if WF(π) = N .

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 3 / 14

Page 14: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Kostant’s theorem

We say π is generic if ∃ψ ∈ Ψ× s.t. Whψ(π) 6= 0.

Theorem (Kostant, Rodier)

π is generic iff it is large.

Theorem (Harish-Chandra, Howe)

Near e ∈ G , the character distribution (asymptotically) equals to a linearcombination of Fourier transforms of Haar measures of nilpotent coadjointorbits.

χπ ≈∑ aOF (µO)

Define WF(π) = ∪{O | aO 6= 0} ⊂ N , where N ⊂ g∗ denotes thenilpotent cone.

π is called large if WF(π) = N .

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 3 / 14

Page 15: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Case of non-generic representations

In the p-adic case, Moeglin and Waldspurger give a very generaldefinition of degenerate Whittaker models and give a preciseconnection between their existence and the wave-front set WF(π).In the real case there is no full analog currently.

Several authors (Matumoto, Yamashita, . . . ) consider generalizedWhittaker functionals ∼ generic characters for smaller nilradicals

We consider degenerate functionals ∼ arbitrary characters of n.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 4 / 14

Page 16: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Case of non-generic representations

In the p-adic case, Moeglin and Waldspurger give a very generaldefinition of degenerate Whittaker models and give a preciseconnection between their existence and the wave-front set WF(π).In the real case there is no full analog currently.

Several authors (Matumoto, Yamashita, . . . ) consider generalizedWhittaker functionals ∼ generic characters for smaller nilradicals

We consider degenerate functionals ∼ arbitrary characters of n.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 4 / 14

Page 17: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Case of non-generic representations

In the p-adic case, Moeglin and Waldspurger give a very generaldefinition of degenerate Whittaker models and give a preciseconnection between their existence and the wave-front set WF(π).In the real case there is no full analog currently.

Several authors (Matumoto, Yamashita, . . . ) consider generalizedWhittaker functionals ∼ generic characters for smaller nilradicals

We consider degenerate functionals ∼ arbitrary characters of n.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 4 / 14

Page 18: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Main results

From now on, let F = R.

The finite group FG = NormGC(G ) / (ZGC

· G ) acts on M (G ).

For π ∈ M (G ) , define π =⊕{πa : a ∈ FG}.

Theorem (1)

For π ∈ M (G ) we have

Ψ(π) ⊂ WF(π) ∩Ψ ⊂ Ψ (π) (1)

Moreover if G = GLn (R) or if G is a complex group then π = π and

Ψ(π) = WF(π) ∩Ψ (2)

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 5 / 14

Page 19: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Main results

From now on, let F = R.

The finite group FG = NormGC(G ) / (ZGC

· G ) acts on M (G ).

For π ∈ M (G ) , define π =⊕{πa : a ∈ FG}.

Theorem (1)

For π ∈ M (G ) we have

Ψ(π) ⊂ WF(π) ∩Ψ ⊂ Ψ (π) (1)

Moreover if G = GLn (R) or if G is a complex group then π = π and

Ψ(π) = WF(π) ∩Ψ (2)

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 5 / 14

Page 20: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Main results

From now on, let F = R.

The finite group FG = NormGC(G ) / (ZGC

· G ) acts on M (G ).

For π ∈ M (G ) , define π =⊕{πa : a ∈ FG}.

Theorem (1)

For π ∈ M (G ) we have

Ψ(π) ⊂ WF(π) ∩Ψ ⊂ Ψ (π) (1)

Moreover if G = GLn (R) or if G is a complex group then π = π and

Ψ(π) = WF(π) ∩Ψ (2)

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 5 / 14

Page 21: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Main results

From now on, let F = R.

The finite group FG = NormGC(G ) / (ZGC

· G ) acts on M (G ).

For π ∈ M (G ) , define π =⊕{πa : a ∈ FG}.

Theorem (1)

For π ∈ M (G ) we have

Ψ(π) ⊂ WF(π) ∩Ψ ⊂ Ψ (π) (1)

Moreover if G = GLn (R) or if G is a complex group then π = π and

Ψ(π) = WF(π) ∩Ψ (2)

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 5 / 14

Page 22: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Main results

Theorem (2)

The sets Ψ (π) and WF (π) determine one another if

1 G = GLn (R) or GLn (C) or SLn (C) and π ∈ M (G )

2 G = Sp2n (C) or On (C) or SOn (C) and π is irreducible

Key observation for the second statement:

Theorem (3)

Let O be a nilpotent orbit for a complex classical Lie algebra then O isuniquely determined by O ∩Ψ.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 6 / 14

Page 23: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Main results

Theorem (2)

The sets Ψ (π) and WF (π) determine one another if

1 G = GLn (R) or GLn (C) or SLn (C) and π ∈ M (G )

2 G = Sp2n (C) or On (C) or SOn (C) and π is irreducible

Key observation for the second statement:

Theorem (3)

Let O be a nilpotent orbit for a complex classical Lie algebra then O isuniquely determined by O ∩Ψ.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 6 / 14

Page 24: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Main results

Theorem (2)

The sets Ψ (π) and WF (π) determine one another if

1 G = GLn (R) or GLn (C) or SLn (C) and π ∈ M (G )

2 G = Sp2n (C) or On (C) or SOn (C) and π is irreducible

Key observation for the second statement:

Theorem (3)

Let O be a nilpotent orbit for a complex classical Lie algebra then O isuniquely determined by O ∩Ψ.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 6 / 14

Page 25: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Main results

Theorem (2)

The sets Ψ (π) and WF (π) determine one another if

1 G = GLn (R) or GLn (C) or SLn (C) and π ∈ M (G )

2 G = Sp2n (C) or On (C) or SOn (C) and π is irreducible

Key observation for the second statement:

Theorem (3)

Let O be a nilpotent orbit for a complex classical Lie algebra then O isuniquely determined by O ∩Ψ.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 6 / 14

Page 26: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Uniqueness

An analog of Shalika’s result would be uniqueness of ”minimallydegenerate” Whittaker models. So far it is known only for GLn, bothin real and p-adic cases.

Let G be GLn(R) or GLn(C). For a partition λ of n let Oλ denotethe corresponding nilpotent orbit and ψλ denote the correspondingcharacter of N.

Theorem (Aizenbud-G-Sahi)

Let π ∈ G be an irreducible unitary representation. Let λ be such thatWF (π) = Oλ. Then dim Whψλ

(π) = 1.

In the p-adic case the analogous theorem was proven by Zelevinskywithout the assumption that π is unitary. The proofs in both casesuse ”derivatives”.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 7 / 14

Page 27: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Uniqueness

An analog of Shalika’s result would be uniqueness of ”minimallydegenerate” Whittaker models. So far it is known only for GLn, bothin real and p-adic cases.

Let G be GLn(R) or GLn(C). For a partition λ of n let Oλ denotethe corresponding nilpotent orbit and ψλ denote the correspondingcharacter of N.

Theorem (Aizenbud-G-Sahi)

Let π ∈ G be an irreducible unitary representation. Let λ be such thatWF (π) = Oλ. Then dim Whψλ

(π) = 1.

In the p-adic case the analogous theorem was proven by Zelevinskywithout the assumption that π is unitary. The proofs in both casesuse ”derivatives”.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 7 / 14

Page 28: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Uniqueness

An analog of Shalika’s result would be uniqueness of ”minimallydegenerate” Whittaker models. So far it is known only for GLn, bothin real and p-adic cases.

Let G be GLn(R) or GLn(C). For a partition λ of n let Oλ denotethe corresponding nilpotent orbit and ψλ denote the correspondingcharacter of N.

Theorem (Aizenbud-G-Sahi)

Let π ∈ G be an irreducible unitary representation. Let λ be such thatWF (π) = Oλ. Then dim Whψλ

(π) = 1.

In the p-adic case the analogous theorem was proven by Zelevinskywithout the assumption that π is unitary. The proofs in both casesuse ”derivatives”.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 7 / 14

Page 29: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Uniqueness

An analog of Shalika’s result would be uniqueness of ”minimallydegenerate” Whittaker models. So far it is known only for GLn, bothin real and p-adic cases.

Let G be GLn(R) or GLn(C). For a partition λ of n let Oλ denotethe corresponding nilpotent orbit and ψλ denote the correspondingcharacter of N.

Theorem (Aizenbud-G-Sahi)

Let π ∈ G be an irreducible unitary representation. Let λ be such thatWF (π) = Oλ. Then dim Whψλ

(π) = 1.

In the p-adic case the analogous theorem was proven by Zelevinskywithout the assumption that π is unitary. The proofs in both casesuse ”derivatives”.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 7 / 14

Page 30: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Algebraic setting

From now on, we let n, g, etc. denote complexified Lie algebras.

Let K ⊂ G be maximal compact subgroup. A (g, K )-module is acomplex vector space with compatible actions of g and K such thatevery vector is K -finite.

Let HC(G ) denote the category of (g, K )-modules of finite length

Theorem (Casselman-Wallach)

The functor π 7→ πK−finite is an equivalence of categoriesM(G ) ∼= HC(G )

For M ∈ HC(G ) and ψ ∈ ΨC we define

Wh′ψ(M) := Homn(M, ψ), Ψ(M) := {ψ ∈ ΨC |Wh′ψ(M) 6= 0}

Kostant showed that π is generic iff πK−finite is generic, thoughdimensions of Whittaker spaces differ considerably.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 8 / 14

Page 31: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Algebraic setting

From now on, we let n, g, etc. denote complexified Lie algebras.

Let K ⊂ G be maximal compact subgroup. A (g, K )-module is acomplex vector space with compatible actions of g and K such thatevery vector is K -finite.

Let HC(G ) denote the category of (g, K )-modules of finite length

Theorem (Casselman-Wallach)

The functor π 7→ πK−finite is an equivalence of categoriesM(G ) ∼= HC(G )

For M ∈ HC(G ) and ψ ∈ ΨC we define

Wh′ψ(M) := Homn(M, ψ), Ψ(M) := {ψ ∈ ΨC |Wh′ψ(M) 6= 0}

Kostant showed that π is generic iff πK−finite is generic, thoughdimensions of Whittaker spaces differ considerably.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 8 / 14

Page 32: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Algebraic setting

From now on, we let n, g, etc. denote complexified Lie algebras.

Let K ⊂ G be maximal compact subgroup. A (g, K )-module is acomplex vector space with compatible actions of g and K such thatevery vector is K -finite.

Let HC(G ) denote the category of (g, K )-modules of finite length

Theorem (Casselman-Wallach)

The functor π 7→ πK−finite is an equivalence of categoriesM(G ) ∼= HC(G )

For M ∈ HC(G ) and ψ ∈ ΨC we define

Wh′ψ(M) := Homn(M, ψ), Ψ(M) := {ψ ∈ ΨC |Wh′ψ(M) 6= 0}

Kostant showed that π is generic iff πK−finite is generic, thoughdimensions of Whittaker spaces differ considerably.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 8 / 14

Page 33: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Algebraic setting

From now on, we let n, g, etc. denote complexified Lie algebras.

Let K ⊂ G be maximal compact subgroup. A (g, K )-module is acomplex vector space with compatible actions of g and K such thatevery vector is K -finite.

Let HC(G ) denote the category of (g, K )-modules of finite length

Theorem (Casselman-Wallach)

The functor π 7→ πK−finite is an equivalence of categoriesM(G ) ∼= HC(G )

For M ∈ HC(G ) and ψ ∈ ΨC we define

Wh′ψ(M) := Homn(M, ψ), Ψ(M) := {ψ ∈ ΨC |Wh′ψ(M) 6= 0}

Kostant showed that π is generic iff πK−finite is generic, thoughdimensions of Whittaker spaces differ considerably.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 8 / 14

Page 34: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Algebraic setting

From now on, we let n, g, etc. denote complexified Lie algebras.

Let K ⊂ G be maximal compact subgroup. A (g, K )-module is acomplex vector space with compatible actions of g and K such thatevery vector is K -finite.

Let HC(G ) denote the category of (g, K )-modules of finite length

Theorem (Casselman-Wallach)

The functor π 7→ πK−finite is an equivalence of categoriesM(G ) ∼= HC(G )

For M ∈ HC(G ) and ψ ∈ ΨC we define

Wh′ψ(M) := Homn(M, ψ), Ψ(M) := {ψ ∈ ΨC |Wh′ψ(M) 6= 0}

Kostant showed that π is generic iff πK−finite is generic, thoughdimensions of Whittaker spaces differ considerably.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 8 / 14

Page 35: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Algebraic setting

From now on, we let n, g, etc. denote complexified Lie algebras.

Let K ⊂ G be maximal compact subgroup. A (g, K )-module is acomplex vector space with compatible actions of g and K such thatevery vector is K -finite.

Let HC(G ) denote the category of (g, K )-modules of finite length

Theorem (Casselman-Wallach)

The functor π 7→ πK−finite is an equivalence of categoriesM(G ) ∼= HC(G )

For M ∈ HC(G ) and ψ ∈ ΨC we define

Wh′ψ(M) := Homn(M, ψ), Ψ(M) := {ψ ∈ ΨC |Wh′ψ(M) 6= 0}

Kostant showed that π is generic iff πK−finite is generic, thoughdimensions of Whittaker spaces differ considerably.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 8 / 14

Page 36: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Associated varieties and our algebraic theorem

Using PBW filtration, grU (g) = Sym(g) = Pol(g∗)

Using this, one can define

AsV(M) ⊂ AnV(M) ⊂ N

Schmid and Vilonen proved that WF(π) and AsV(πK−finite)determine each other.

Let prn∗ : g∗ → n∗ denote the natural projection (restriction to n).

Theorem (0)

For M ∈ HC we have Ψ(M) = prn∗(AsV (M)) ∩Ψ.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 9 / 14

Page 37: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Associated varieties and our algebraic theorem

Using PBW filtration, grU (g) = Sym(g) = Pol(g∗)

Using this, one can define

AsV(M) ⊂ AnV(M) ⊂ N

Schmid and Vilonen proved that WF(π) and AsV(πK−finite)determine each other.

Let prn∗ : g∗ → n∗ denote the natural projection (restriction to n).

Theorem (0)

For M ∈ HC we have Ψ(M) = prn∗(AsV (M)) ∩Ψ.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 9 / 14

Page 38: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Associated varieties and our algebraic theorem

Using PBW filtration, grU (g) = Sym(g) = Pol(g∗)

Using this, one can define

AsV(M) ⊂ AnV(M) ⊂ N

Schmid and Vilonen proved that WF(π) and AsV(πK−finite)determine each other.

Let prn∗ : g∗ → n∗ denote the natural projection (restriction to n).

Theorem (0)

For M ∈ HC we have Ψ(M) = prn∗(AsV (M)) ∩Ψ.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 9 / 14

Page 39: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Associated varieties and our algebraic theorem

Using PBW filtration, grU (g) = Sym(g) = Pol(g∗)

Using this, one can define

AsV(M) ⊂ AnV(M) ⊂ N

Schmid and Vilonen proved that WF(π) and AsV(πK−finite)determine each other.

Let prn∗ : g∗ → n∗ denote the natural projection (restriction to n).

Theorem (0)

For M ∈ HC we have Ψ(M) = prn∗(AsV (M)) ∩Ψ.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 9 / 14

Page 40: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Associated varieties and our algebraic theorem

Using PBW filtration, grU (g) = Sym(g) = Pol(g∗)

Using this, one can define

AsV(M) ⊂ AnV(M) ⊂ N

Schmid and Vilonen proved that WF(π) and AsV(πK−finite)determine each other.

Let prn∗ : g∗ → n∗ denote the natural projection (restriction to n).

Theorem (0)

For M ∈ HC we have Ψ(M) = prn∗(AsV (M)) ∩Ψ.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 9 / 14

Page 41: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Idea of the proof

Since n/[n, n] is commutative, from Nakayama’s lemma we haveΨ(M) = Supp(M/[n, n]M). Now, restriction to n corresponds toprojection on n∗ and quotient by [n, n] corresponds to intersectionwith Ψ = [n, n]⊥.

However, in non-commutative situation one could even haveV = [n, n]V . For example, let G = GL(3, R) and consider theidentification of n with the Heisenberg Lie algebra

⟨x , d

dx , 1⟩

actingon V = C [x ].

Let b = h+ n be the Borel subalgebra of g, let V be a b-module. We

define the n-adic completion and Jacquet module as follows:

V = Vn = lim←−

V /ni V , J (V ) = Jb (V ) =(

Vn

)h-finite

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 10 / 14

Page 42: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Idea of the proof

Since n/[n, n] is commutative, from Nakayama’s lemma we haveΨ(M) = Supp(M/[n, n]M). Now, restriction to n corresponds toprojection on n∗ and quotient by [n, n] corresponds to intersectionwith Ψ = [n, n]⊥.

However, in non-commutative situation one could even haveV = [n, n]V . For example, let G = GL(3, R) and consider theidentification of n with the Heisenberg Lie algebra

⟨x , d

dx , 1⟩

actingon V = C [x ].

Let b = h+ n be the Borel subalgebra of g, let V be a b-module. We

define the n-adic completion and Jacquet module as follows:

V = Vn = lim←−

V /ni V , J (V ) = Jb (V ) =(

Vn

)h-finite

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 10 / 14

Page 43: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Idea of the proof

Since n/[n, n] is commutative, from Nakayama’s lemma we haveΨ(M) = Supp(M/[n, n]M). Now, restriction to n corresponds toprojection on n∗ and quotient by [n, n] corresponds to intersectionwith Ψ = [n, n]⊥.

However, in non-commutative situation one could even haveV = [n, n]V . For example, let G = GL(3, R) and consider theidentification of n with the Heisenberg Lie algebra

⟨x , d

dx , 1⟩

actingon V = C [x ].

Let b = h+ n be the Borel subalgebra of g, let V be a b-module. We

define the n-adic completion and Jacquet module as follows:

V = Vn = lim←−

V /ni V , J (V ) = Jb (V ) =(

Vn

)h-finite

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 10 / 14

Page 44: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Sketch of the proof

Define n′ = [n, n] and CV = H0 (n′, V ) = V /n′V .

(Nakayama) Ψ(M) = Suppv(CM) = AnVv (CM)

(Joseph+Gabber) AnVv (CM) = AnVv(

CM)= AnVv (J (CM))

(Easy) J(CM) ≈ C (JM) as b-modules.

(Bernstein+Joseph)AnVv (J (CM)) = AsVv(C (JM)) = AsVn(JM) ∩Ψ.

(Ginzburg+ENV) AsVn(JM) ⊃ AsVn(M) ∩Ψ.

(Casselman-Osborne+Gabber) AsVn(M) = prn∗(AsVg(M)).

Thus Ψ(M) ⊃ prn∗(AsVg(M)) ∩Ψ; other inclusion is easy.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 11 / 14

Page 45: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Sketch of the proof

Define n′ = [n, n] and CV = H0 (n′, V ) = V /n′V .

(Nakayama) Ψ(M) = Suppv(CM) = AnVv (CM)

(Joseph+Gabber) AnVv (CM) = AnVv(

CM)= AnVv (J (CM))

(Easy) J(CM) ≈ C (JM) as b-modules.

(Bernstein+Joseph)AnVv (J (CM)) = AsVv(C (JM)) = AsVn(JM) ∩Ψ.

(Ginzburg+ENV) AsVn(JM) ⊃ AsVn(M) ∩Ψ.

(Casselman-Osborne+Gabber) AsVn(M) = prn∗(AsVg(M)).

Thus Ψ(M) ⊃ prn∗(AsVg(M)) ∩Ψ; other inclusion is easy.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 11 / 14

Page 46: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Sketch of the proof

Define n′ = [n, n] and CV = H0 (n′, V ) = V /n′V .

(Nakayama) Ψ(M) = Suppv(CM) = AnVv (CM)

(Joseph+Gabber) AnVv (CM) = AnVv(

CM)= AnVv (J (CM))

(Easy) J(CM) ≈ C (JM) as b-modules.

(Bernstein+Joseph)AnVv (J (CM)) = AsVv(C (JM)) = AsVn(JM) ∩Ψ.

(Ginzburg+ENV) AsVn(JM) ⊃ AsVn(M) ∩Ψ.

(Casselman-Osborne+Gabber) AsVn(M) = prn∗(AsVg(M)).

Thus Ψ(M) ⊃ prn∗(AsVg(M)) ∩Ψ; other inclusion is easy.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 11 / 14

Page 47: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Sketch of the proof

Define n′ = [n, n] and CV = H0 (n′, V ) = V /n′V .

(Nakayama) Ψ(M) = Suppv(CM) = AnVv (CM)

(Joseph+Gabber) AnVv (CM) = AnVv(

CM)= AnVv (J (CM))

(Easy) J(CM) ≈ C (JM) as b-modules.

(Bernstein+Joseph)AnVv (J (CM)) = AsVv(C (JM)) = AsVn(JM) ∩Ψ.

(Ginzburg+ENV) AsVn(JM) ⊃ AsVn(M) ∩Ψ.

(Casselman-Osborne+Gabber) AsVn(M) = prn∗(AsVg(M)).

Thus Ψ(M) ⊃ prn∗(AsVg(M)) ∩Ψ; other inclusion is easy.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 11 / 14

Page 48: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Sketch of the proof

Define n′ = [n, n] and CV = H0 (n′, V ) = V /n′V .

(Nakayama) Ψ(M) = Suppv(CM) = AnVv (CM)

(Joseph+Gabber) AnVv (CM) = AnVv(

CM)= AnVv (J (CM))

(Easy) J(CM) ≈ C (JM) as b-modules.

(Bernstein+Joseph)AnVv (J (CM)) = AsVv(C (JM)) = AsVn(JM) ∩Ψ.

(Ginzburg+ENV) AsVn(JM) ⊃ AsVn(M) ∩Ψ.

(Casselman-Osborne+Gabber) AsVn(M) = prn∗(AsVg(M)).

Thus Ψ(M) ⊃ prn∗(AsVg(M)) ∩Ψ; other inclusion is easy.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 11 / 14

Page 49: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Sketch of the proof

Define n′ = [n, n] and CV = H0 (n′, V ) = V /n′V .

(Nakayama) Ψ(M) = Suppv(CM) = AnVv (CM)

(Joseph+Gabber) AnVv (CM) = AnVv(

CM)= AnVv (J (CM))

(Easy) J(CM) ≈ C (JM) as b-modules.

(Bernstein+Joseph)AnVv (J (CM)) = AsVv(C (JM)) = AsVn(JM) ∩Ψ.

(Ginzburg+ENV) AsVn(JM) ⊃ AsVn(M) ∩Ψ.

(Casselman-Osborne+Gabber) AsVn(M) = prn∗(AsVg(M)).

Thus Ψ(M) ⊃ prn∗(AsVg(M)) ∩Ψ; other inclusion is easy.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 11 / 14

Page 50: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Sketch of the proof

Define n′ = [n, n] and CV = H0 (n′, V ) = V /n′V .

(Nakayama) Ψ(M) = Suppv(CM) = AnVv (CM)

(Joseph+Gabber) AnVv (CM) = AnVv(

CM)= AnVv (J (CM))

(Easy) J(CM) ≈ C (JM) as b-modules.

(Bernstein+Joseph)AnVv (J (CM)) = AsVv(C (JM)) = AsVn(JM) ∩Ψ.

(Ginzburg+ENV) AsVn(JM) ⊃ AsVn(M) ∩Ψ.

(Casselman-Osborne+Gabber) AsVn(M) = prn∗(AsVg(M)).

Thus Ψ(M) ⊃ prn∗(AsVg(M)) ∩Ψ; other inclusion is easy.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 11 / 14

Page 51: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Sketch of the proof

Define n′ = [n, n] and CV = H0 (n′, V ) = V /n′V .

(Nakayama) Ψ(M) = Suppv(CM) = AnVv (CM)

(Joseph+Gabber) AnVv (CM) = AnVv(

CM)= AnVv (J (CM))

(Easy) J(CM) ≈ C (JM) as b-modules.

(Bernstein+Joseph)AnVv (J (CM)) = AsVv(C (JM)) = AsVn(JM) ∩Ψ.

(Ginzburg+ENV) AsVn(JM) ⊃ AsVn(M) ∩Ψ.

(Casselman-Osborne+Gabber) AsVn(M) = prn∗(AsVg(M)).

Thus Ψ(M) ⊃ prn∗(AsVg(M)) ∩Ψ; other inclusion is easy.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 11 / 14

Page 52: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Proof of Theorem 3

Proof.

For GL (n, R) and SL (n, C) ∼ Jordan form

Orbits for Sp2n (C) or On (C) ∼ partitions satisfying certainconditions

An orbit meets Ψ iff it has at most one part ≥ 2 with odd multiplicity

For each partition λ and each k there is a partition µ ≤ λ, whichmeets Ψ and satisfies µ1 + · · ·+ µk = λ1 + · · ·+ λk

Result for SOn (C) requires slight additional argument.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 12 / 14

Page 53: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Proof of Theorem 3

Proof.

For GL (n, R) and SL (n, C) ∼ Jordan form

Orbits for Sp2n (C) or On (C) ∼ partitions satisfying certainconditions

An orbit meets Ψ iff it has at most one part ≥ 2 with odd multiplicity

For each partition λ and each k there is a partition µ ≤ λ, whichmeets Ψ and satisfies µ1 + · · ·+ µk = λ1 + · · ·+ λk

Result for SOn (C) requires slight additional argument.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 12 / 14

Page 54: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Proof of Theorem 3

Proof.

For GL (n, R) and SL (n, C) ∼ Jordan form

Orbits for Sp2n (C) or On (C) ∼ partitions satisfying certainconditions

An orbit meets Ψ iff it has at most one part ≥ 2 with odd multiplicity

For each partition λ and each k there is a partition µ ≤ λ, whichmeets Ψ and satisfies µ1 + · · ·+ µk = λ1 + · · ·+ λk

Result for SOn (C) requires slight additional argument.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 12 / 14

Page 55: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Proof of Theorem 3

Proof.

For GL (n, R) and SL (n, C) ∼ Jordan form

Orbits for Sp2n (C) or On (C) ∼ partitions satisfying certainconditions

An orbit meets Ψ iff it has at most one part ≥ 2 with odd multiplicity

For each partition λ and each k there is a partition µ ≤ λ, whichmeets Ψ and satisfies µ1 + · · ·+ µk = λ1 + · · ·+ λk

Result for SOn (C) requires slight additional argument.

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 12 / 14

Page 56: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Counterexamples for exceptional groups

Fact

Theorem 3 is false for every exceptional group.

We list all orbits whose closures have the same intersection with Ψ.

We follow Bala-Carter notation and we have underlined the special orbits.

For G = G2: G2(a1) and A1

For G = F4:

1 F4(a1) and F4(a2)2 F4(a3) and C3(a1)

For G = E6:

1 E6(a1) and D5

2 D4(a1) and A3 + A1

For G = E7:

1 E7(a1) and E7(a2)2 E7(a3) and D6

3 E6(a1) and E7(a4).

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 13 / 14

Page 57: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Counterexamples for exceptional groups

Fact

Theorem 3 is false for every exceptional group.

We list all orbits whose closures have the same intersection with Ψ.

We follow Bala-Carter notation and we have underlined the special orbits.

For G = G2: G2(a1) and A1

For G = F4:

1 F4(a1) and F4(a2)2 F4(a3) and C3(a1)

For G = E6:

1 E6(a1) and D5

2 D4(a1) and A3 + A1

For G = E7:

1 E7(a1) and E7(a2)2 E7(a3) and D6

3 E6(a1) and E7(a4).

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 13 / 14

Page 58: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Counterexamples for exceptional groups

Fact

Theorem 3 is false for every exceptional group.

We list all orbits whose closures have the same intersection with Ψ.

We follow Bala-Carter notation and we have underlined the special orbits.

For G = G2: G2(a1) and A1

For G = F4:

1 F4(a1) and F4(a2)2 F4(a3) and C3(a1)

For G = E6:

1 E6(a1) and D5

2 D4(a1) and A3 + A1

For G = E7:

1 E7(a1) and E7(a2)2 E7(a3) and D6

3 E6(a1) and E7(a4).

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 13 / 14

Page 59: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Counterexamples for exceptional groups

Fact

Theorem 3 is false for every exceptional group.

We list all orbits whose closures have the same intersection with Ψ.

We follow Bala-Carter notation and we have underlined the special orbits.

For G = G2: G2(a1) and A1

For G = F4:

1 F4(a1) and F4(a2)2 F4(a3) and C3(a1)

For G = E6:

1 E6(a1) and D5

2 D4(a1) and A3 + A1

For G = E7:

1 E7(a1) and E7(a2)2 E7(a3) and D6

3 E6(a1) and E7(a4).

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 13 / 14

Page 60: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Counterexamples for exceptional groups

Fact

Theorem 3 is false for every exceptional group.

We list all orbits whose closures have the same intersection with Ψ.

We follow Bala-Carter notation and we have underlined the special orbits.

For G = G2: G2(a1) and A1

For G = F4:

1 F4(a1) and F4(a2)2 F4(a3) and C3(a1)

For G = E6:

1 E6(a1) and D5

2 D4(a1) and A3 + A1

For G = E7:

1 E7(a1) and E7(a2)2 E7(a3) and D6

3 E6(a1) and E7(a4).

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 13 / 14

Page 61: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Counterexamples for exceptional groups

Fact

Theorem 3 is false for every exceptional group.

We list all orbits whose closures have the same intersection with Ψ.

We follow Bala-Carter notation and we have underlined the special orbits.

For G = G2: G2(a1) and A1

For G = F4:1 F4(a1) and F4(a2)

2 F4(a3) and C3(a1)

For G = E6:

1 E6(a1) and D5

2 D4(a1) and A3 + A1

For G = E7:

1 E7(a1) and E7(a2)2 E7(a3) and D6

3 E6(a1) and E7(a4).

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 13 / 14

Page 62: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Counterexamples for exceptional groups

Fact

Theorem 3 is false for every exceptional group.

We list all orbits whose closures have the same intersection with Ψ.

We follow Bala-Carter notation and we have underlined the special orbits.

For G = G2: G2(a1) and A1

For G = F4:1 F4(a1) and F4(a2)2 F4(a3) and C3(a1)

For G = E6:

1 E6(a1) and D5

2 D4(a1) and A3 + A1

For G = E7:

1 E7(a1) and E7(a2)2 E7(a3) and D6

3 E6(a1) and E7(a4).

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 13 / 14

Page 63: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Counterexamples for exceptional groups

Fact

Theorem 3 is false for every exceptional group.

We list all orbits whose closures have the same intersection with Ψ.

We follow Bala-Carter notation and we have underlined the special orbits.

For G = G2: G2(a1) and A1

For G = F4:1 F4(a1) and F4(a2)2 F4(a3) and C3(a1)

For G = E6:

1 E6(a1) and D5

2 D4(a1) and A3 + A1

For G = E7:

1 E7(a1) and E7(a2)2 E7(a3) and D6

3 E6(a1) and E7(a4).

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 13 / 14

Page 64: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Counterexamples for exceptional groups

Fact

Theorem 3 is false for every exceptional group.

We list all orbits whose closures have the same intersection with Ψ.

We follow Bala-Carter notation and we have underlined the special orbits.

For G = G2: G2(a1) and A1

For G = F4:1 F4(a1) and F4(a2)2 F4(a3) and C3(a1)

For G = E6:1 E6(a1) and D5

2 D4(a1) and A3 + A1

For G = E7:

1 E7(a1) and E7(a2)2 E7(a3) and D6

3 E6(a1) and E7(a4).

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 13 / 14

Page 65: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Counterexamples for exceptional groups

Fact

Theorem 3 is false for every exceptional group.

We list all orbits whose closures have the same intersection with Ψ.

We follow Bala-Carter notation and we have underlined the special orbits.

For G = G2: G2(a1) and A1

For G = F4:1 F4(a1) and F4(a2)2 F4(a3) and C3(a1)

For G = E6:1 E6(a1) and D5

2 D4(a1) and A3 + A1

For G = E7:

1 E7(a1) and E7(a2)2 E7(a3) and D6

3 E6(a1) and E7(a4).

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 13 / 14

Page 66: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Counterexamples for exceptional groups

Fact

Theorem 3 is false for every exceptional group.

We list all orbits whose closures have the same intersection with Ψ.

We follow Bala-Carter notation and we have underlined the special orbits.

For G = G2: G2(a1) and A1

For G = F4:1 F4(a1) and F4(a2)2 F4(a3) and C3(a1)

For G = E6:1 E6(a1) and D5

2 D4(a1) and A3 + A1

For G = E7:

1 E7(a1) and E7(a2)2 E7(a3) and D6

3 E6(a1) and E7(a4).

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 13 / 14

Page 67: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Counterexamples for exceptional groups

Fact

Theorem 3 is false for every exceptional group.

We list all orbits whose closures have the same intersection with Ψ.

We follow Bala-Carter notation and we have underlined the special orbits.

For G = G2: G2(a1) and A1

For G = F4:1 F4(a1) and F4(a2)2 F4(a3) and C3(a1)

For G = E6:1 E6(a1) and D5

2 D4(a1) and A3 + A1

For G = E7:1 E7(a1) and E7(a2)

2 E7(a3) and D6

3 E6(a1) and E7(a4).

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 13 / 14

Page 68: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Counterexamples for exceptional groups

Fact

Theorem 3 is false for every exceptional group.

We list all orbits whose closures have the same intersection with Ψ.

We follow Bala-Carter notation and we have underlined the special orbits.

For G = G2: G2(a1) and A1

For G = F4:1 F4(a1) and F4(a2)2 F4(a3) and C3(a1)

For G = E6:1 E6(a1) and D5

2 D4(a1) and A3 + A1

For G = E7:1 E7(a1) and E7(a2)2 E7(a3) and D6

3 E6(a1) and E7(a4).

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 13 / 14

Page 69: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Counterexamples for exceptional groups

Fact

Theorem 3 is false for every exceptional group.

We list all orbits whose closures have the same intersection with Ψ.

We follow Bala-Carter notation and we have underlined the special orbits.

For G = G2: G2(a1) and A1

For G = F4:1 F4(a1) and F4(a2)2 F4(a3) and C3(a1)

For G = E6:1 E6(a1) and D5

2 D4(a1) and A3 + A1

For G = E7:1 E7(a1) and E7(a2)2 E7(a3) and D6

3 E6(a1) and E7(a4).

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 13 / 14

Page 70: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Counterexamples for exceptional groups

For G = E8:

1 E8(a1), E8(a2), and E8(a3)2 E8(a4), E8(b4) and E8(a5)3 E7(a1), E8(b5) and E7(a2)4 E8(a6) and D7(a1)5 E6(a1) and E7(a4)6 E8(a7), E7(a5), E6(a3) + A1, and D6(a2).

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 14 / 14

Page 71: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Counterexamples for exceptional groups

For G = E8:

1 E8(a1), E8(a2), and E8(a3)

2 E8(a4), E8(b4) and E8(a5)3 E7(a1), E8(b5) and E7(a2)4 E8(a6) and D7(a1)5 E6(a1) and E7(a4)6 E8(a7), E7(a5), E6(a3) + A1, and D6(a2).

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 14 / 14

Page 72: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Counterexamples for exceptional groups

For G = E8:

1 E8(a1), E8(a2), and E8(a3)2 E8(a4), E8(b4) and E8(a5)

3 E7(a1), E8(b5) and E7(a2)4 E8(a6) and D7(a1)5 E6(a1) and E7(a4)6 E8(a7), E7(a5), E6(a3) + A1, and D6(a2).

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 14 / 14

Page 73: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Counterexamples for exceptional groups

For G = E8:

1 E8(a1), E8(a2), and E8(a3)2 E8(a4), E8(b4) and E8(a5)3 E7(a1), E8(b5) and E7(a2)

4 E8(a6) and D7(a1)5 E6(a1) and E7(a4)6 E8(a7), E7(a5), E6(a3) + A1, and D6(a2).

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 14 / 14

Page 74: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Counterexamples for exceptional groups

For G = E8:

1 E8(a1), E8(a2), and E8(a3)2 E8(a4), E8(b4) and E8(a5)3 E7(a1), E8(b5) and E7(a2)4 E8(a6) and D7(a1)

5 E6(a1) and E7(a4)6 E8(a7), E7(a5), E6(a3) + A1, and D6(a2).

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 14 / 14

Page 75: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Counterexamples for exceptional groups

For G = E8:

1 E8(a1), E8(a2), and E8(a3)2 E8(a4), E8(b4) and E8(a5)3 E7(a1), E8(b5) and E7(a2)4 E8(a6) and D7(a1)5 E6(a1) and E7(a4)

6 E8(a7), E7(a5), E6(a3) + A1, and D6(a2).

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 14 / 14

Page 76: Degenerate Whittaker functionals for real reductive groupsdimagur/DimaTalkDegWhit.pdf · Degenerate Whittaker functionals for real reductive groups Dmitry Gourevitch & Siddhartha

Counterexamples for exceptional groups

For G = E8:

1 E8(a1), E8(a2), and E8(a3)2 E8(a4), E8(b4) and E8(a5)3 E7(a1), E8(b5) and E7(a2)4 E8(a6) and D7(a1)5 E6(a1) and E7(a4)6 E8(a7), E7(a5), E6(a3) + A1, and D6(a2).

Gourevitch-Sahi Degenerate Whittaker functionals August 2012 14 / 14