1 / 10

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.

kblack
Download Presentation

Multiplicity one theorems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related