1. 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:0709.4215 [math.RT] Let F be a non-archimedean local field of characteristic zero.

2. 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.

3. Localization principle

4. Frobenius reciprocity

5. 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.

6. Geometric Symmetries

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

8. 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

9. Fourier transform & Homogeneity lemma

10. 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