Degenerate Whittaker functionals for real reductive groups

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Associated varieties and our algebraic theorem Using PBW filtration, gr U(g) = Sym(g) = Pol(g ∗ ) Using this, one can define AsV(M) ⊂ AnV(M) ⊂ N Schmid and Vilonen proved that WF(π) and AsV(π K −finite ) determine each other. Let pr n ∗ : g ∗ → n ∗ denote the natural projection (restriction to n). Theorem (0) For M ∈ HC we have Ψ(M) = pr n ∗(AsV (M)) ∩ Ψ. Gourevitch-Sahi Degenerate Whittaker functionals August 2012 9 / 14
Idea of the proof Since n/[n, n] is commutative, from Nakayama’s lemma we have Ψ(M) = Supp(M/[n, n]M). Now, restriction to n corresponds to projection on n ∗ and quotient by [n, n] corresponds to intersection with Ψ = [n, n] ⊥ . Gourevitch-Sahi Degenerate Whittaker functionals August 2012 10 / 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 and 28: Uniqueness An analog of Shalika’s
Page 29 and 30: 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: Associated varieties and our algebr
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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