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Multiplicity one theorems

Multiplicity one theorems. A. Aizenbud, D. Gourevitch S. Rallis and G. Schiffmann. arXiv:0709.4215 [math.RT]. Let F be a non-archimedean local field of characteristic zero. Theorem A Every GL( n; F) invariant distribution on GL( n + 1 ; F) is invariant with respect to transposition.

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Multiplicity one theorems

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  1. Multiplicity one theorems A. Aizenbud, D. Gourevitch S. Rallis and G. Schiffmann arXiv:0709.4215 [math.RT] Let F be a non-archimedean local field of characteristic zero. Theorem AEvery GL(n; F) invariant distribution on GL(n + 1; F) is invariant with respect to transposition. it implies Theorem BLet p be an irreducible smooth representation of GL(n + 1; F) and let r be an irreducible smooth representation of GL(n; F). Then Theorem B2Let p be an irreducible smooth representation of O(n + 1; F) and let r be an irreducible smooth representation of O(n; F). Then

  2. 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. 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*(Si)G=0. Then S*(X)G=0.

  3. Localization principle

  4. Frobenius reciprocity

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

  6. Geometric Symmetries

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

  8. Fourier transform & Homogeneity lemma

  9. 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. LetW:=V⊕D. Extend < , > to W in the obvious way. Consider the embedding of U(V) into U(W). Theorem A2Every U(V)- invariant distribution on U(W) is invariant with respect totransposition. it implies Theorem B2Let p be an irreducible smooth representation of U(W) and let r be an irreducible smooth representation of U(V).Then

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