Multiplicity one theorems A. Aizenbud, D. Gourevitch S. Rallis and G. Schiffmann Theorem A Every GL(n; F) invariant distribution on GL(n + 1; F) is invariant.

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Presentation transcript:

Multiplicity one theorems A. Aizenbud, D. Gourevitch S. Rallis and G. Schiffmann Theorem A Every GL(n; F) invariant distribution on GL(n + 1; F) is invariant with respect to transposition. Theorem B Let  be an irreducible smooth representation of GL(n + 1; F) and let  be an irreducible smooth representation of GL(n; F). Then it implies Theorem B 2 Let  be an irreducible smooth representation of O(n + 1; F) and let  be an irreducible smooth representation of O(n; F). Then arXiv: [math.RT] Let F be a non-archimedean local field of characteristic zero.

Corollary. Let an l-group G act on an l-space X. Let be a finite G-invariant stratification. Suppose that for any i, S*(S i ) G =0. Then S*(X) G =0. Let X be an l-space (i.e. Hausdorff locally compact totally disconnected topological space). Denote by S(X) the space of locally constant compactly supported functions. Denote also S*(X):=(S(X))* For closed subset Z of X, 0 → S*(Z) → S*(X) → S*(X\Z) →0.

Localization principle

Frobenius reciprocity

Proof Reformulation: Proof of Gelfand-Kazhdan Theorem Every fiber has finite number of orbits For every orbit we use Frobenius reciprocity and the fact that A and A t are conjugate. Here, q is the “characteristic polynomial” map, and P is the space of monic polynomials of degree n. Localization principle Theorem (Gelfand-Kazhdan). Every GL(n ; F) invariant distribution on GL(n ; F) is invariant with respect to transposition.

Geometric Symmetries

Fourier transform Homogeneity lemma The proof of this lemma uses Weil representation.

ToolUsed inUsed by Bernstein’s localization principle (GL n, GL n ) Gel’fand and Kazhdan Frobenius reciprocity Fourier transform(P n, GL n )Bernstein Weil representation (GL n xGL k, GL n+k ) two-sided action Jacquet and Rallis (GL n, GL n+1 ) Rallis and Schiffmann Geometric Symmetries Aizenbud and Gourevitch

Fourier transform & Homogeneity lemma

Let D be either F or a quadratic extension of F. Let V be a vector space over D of dimension n. Let be a non-degenerate hermitian form on V. Let W:=V⊕D. Extend to W in the obvious way. Consider the embedding of U(V) into U(W). Theorem A 2 Every U(V)- invariant distribution on U(W) is invariant with respect to transposition. it implies Theorem B 2 Let  be an irreducible smooth representation of U(W) and let  be an irreducible smooth representation of U(V). Then