Degenerate Whittaker functionals for real reductive groups

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Uniqueness An analog of Shalika’s result would be uniqueness of ”minimally degenerate” Whittaker models. So far it is known only for GL n , both in real and p-adic cases. Let G be GL n (R) or GL n (C). For a partition λ of n let O λ denote the corresponding nilpotent orbit and ψ λ denote the corresponding character of N. Theorem (Aizenbud-G-Sahi) Let π ∈ Ĝ be an irreducible unitary representation. Let λ be such that WF (π) = O λ . Then dim Wh ψλ (π) = 1. Gourevitch-Sahi Degenerate Whittaker functionals August 2012 7 / 14
Uniqueness An analog of Shalika’s result would be uniqueness of ”minimally degenerate” Whittaker models. So far it is known only for GL n , both in real and p-adic cases. Let G be GL n (R) or GL n (C). For a partition λ of n let O λ denote the corresponding nilpotent orbit and ψ λ denote the corresponding character of N. Theorem (Aizenbud-G-Sahi) Let π ∈ Ĝ be an irreducible unitary representation. Let λ be such that WF (π) = O λ . Then dim Wh ψλ (π) = 1. In the p-adic case the analogous theorem was proven by Zelevinsky without the assumption that π is unitary. The proofs in both cases use ”derivatives”. Gourevitch-Sahi Degenerate Whittaker functionals August 2012 7 / 14
Page 1 and 2: Degenerate Whittaker functionals fo
Page 3 and 4: Whittaker functionals F: local fiel
Page 11 and 12: Kostant’s theorem We say π is ge
Page 13 and 14: Kostant’s theorem We say π is ge
Page 15 and 16: Case of non-generic representations
Page 17 and 18: Case of non-generic representations
Page 19 and 20: Main results From now on, let F = R
Page 21 and 22: Main results From now on, let F = R
Page 23 and 24: Main results Theorem (2) The sets
Page 25 and 26: Main results Theorem (2) The sets
Page 27: Uniqueness An analog of Shalika’s
Page 31 and 32: Algebraic setting From now on, we l
Page 37 and 38: Associated varieties and our algebr
Page 39 and 40: Associated varieties and our algebr
Page 41 and 42: Idea of the proof Since n/[n, n] is
Page 43 and 44: Idea of the proof Since n/[n, n] is
Page 45 and 46: Sketch of the proof Define n ′ =
Page 53 and 54: Proof of Theorem 3 Proof. For GL (n
Page 55 and 56: Proof of Theorem 3 Proof. For GL (n
Page 57 and 58: Counterexamples for exceptional gro

Degenerate Whittaker functionals for real reductive groups

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