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.

Slides:



Advertisements
Similar presentations
Vector Spaces A set V is called a vector space over a set K denoted V(K) if is an Abelian group, is a field, and For every element vV and K there exists.
Advertisements

8.3 Inverse Linear Transformations
Singularity of a Holomorphic Map Jing Zhang State University of New York, Albany.
Finite Fields Rong-Jaye Chen. p2. Finite fields 1. Irreducible polynomial f(x)  K[x], f(x) has no proper divisors in K[x] Eg. f(x)=1+x+x 2 is irreducible.
Shuijing Crystal Li Rice University Mathematics Department 1 Rational Points on del Pezzo Surfaces of degree 1 and 2.
Degenerations of algebras. José-Antonio de la Peña UNAM, México Advanced School and Conference on Homological and Geometrical Methods in Representation.
Curves and Surfaces from 3-D Matrices Dan Dreibelbis University of North Florida.
Probabilistic verification Mario Szegedy, Rutgers www/cs.rutgers.edu/~szegedy/07540 Lecture 3.
Information and Coding Theory Finite fields. Juris Viksna, 2015.
Chaper 3 Weak Topologies. Reflexive Space.Separabe Space. Uniform Convex Spaces.
THE DIMENSION OF A VECTOR SPACE
Chapter 5 Orthogonality
On the Toric Varieties Associated with Bicolored Metric Trees Khoa Lu Nguyen Joseph Shao Summer REU 2008 at Rutgers University Advisors: Prof. Woodward.
6. One-Dimensional Continuous Groups 6.1 The Rotation Group SO(2) 6.2 The Generator of SO(2) 6.3 Irreducible Representations of SO(2) 6.4 Invariant Integration.
Correcting Errors Beyond the Guruswami-Sudan Radius Farzad Parvaresh & Alexander Vardy Presented by Efrat Bank.
Introduction Polynomials
Finite fields.
5. Similarity I.Complex Vector Spaces II.Similarity III.Nilpotence IV.Jordan Form Topics Goal: Given H = h B → B, find D s.t. K = h D → D has a simple.
CS Subdivision I: The Univariate Setting Peter Schröder.
Cardinality of a Set “The number of elements in a set.” Let A be a set. a.If A =  (the empty set), then the cardinality of A is 0. b. If A has exactly.
Manindra Agrawal NUS / IITK
Maximum Likelihood Estimation
CHAPTER SIX Eigenvalues
Preperiodic Points and Unlikely Intersections joint work with Laura DeMarco Matthew Baker Georgia Institute of Technology AMS Southeastern Section Meeting.
GROUPS & THEIR REPRESENTATIONS: a card shuffling approach Wayne Lawton Department of Mathematics National University of Singapore S ,
Integrable hierarchies of
Chapter 2: Vector spaces
6.3 Permutation groups and cyclic groups  Example: Consider the equilateral triangle with vertices 1 , 2 , and 3. Let l 1, l 2, and l 3 be the angle bisectors.
Chap. 6 Linear Transformations
HERMITE INTERPOLATION in LOOP GROUPS and CONJUGATE QUADRATURE FILTER APPROXIMATION Wayne Lawton Department of Mathematics National University of Singapore.
Three different ways There are three different ways to show that ρ(A) is a simple eigenvalue of an irreducible nonnegative matrix A:
Signal Processing and Representation Theory Lecture 2.
3.4 Zeros of Polynomial Functions. The Fundamental Theorem of Algebra If f(x) is a polynomial of degree n, where n>0, then f has at least one zero in.
A. Aizenbud and D. Gourevitch Gelfand Pairs Regular pairs We call the property (2)
March 24, 2006 Fixed Point Theory in Fréchet Space D. P. Dwiggins Systems Support Office of Admissions Department of Mathematical Sciences Analysis Seminar.
Chapter 4 Hilbert Space. 4.1 Inner product space.
Chap. 4 Vector Spaces 4.1 Vectors in Rn 4.2 Vector Spaces
Positively Expansive Maps and Resolution of Singularities Wayne Lawton Department of Mathematics National University of Singapore
Positive Semidefinite matrix A is a p ositive semidefinite matrix (also called n onnegative definite matrix)
Projective Geometry Hu Zhan Yi. Entities At Infinity The ordinary space in which we lie is Euclidean space. The parallel lines usually do not intersect.
2.5 The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra The Fundamental Theorem of Algebra – If f(x) is a polynomial of degree n, where.
Section 6: Fundamental Theorem of Algebra Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with.
SECTION 8 Groups of Permutations Definition A permutation of a set A is a function  ϕ : A  A that is both one to one and onto. If  and  are both permutations.
Conjugate Pairs Theorem Every complex polynomial function of degree n  1 has exactly n complex zeros, some of which may repeat. 1) A polynomial function.
4 4.5 © 2016 Pearson Education, Inc. Vector Spaces THE DIMENSION OF A VECTOR SPACE.
ALGEBRAIC CURVES AND CONTROL THEORY by Bill Wolovich Based on Chapter 3 of the book INVARIANTS FOR PATTERN RECOGNITION AND CLASSIFICATION, World Scientific.
Extending a displacement A displacement defined by a pair where l is the length of the displacement and  the angle between its direction and the x-axix.
Lecture from Quantum Mechanics. Marek Zrałek Field Theory and Particle Physics Department. Silesian University Lecture 6.
Beyond Vectors Hung-yi Lee. Introduction Many things can be considered as “vectors”. E.g. a function can be regarded as a vector We can apply the concept.
Functions of Complex Variable and Integral Transforms
Mathematical Background: Extension Fields
Theorem of Banach stainhaus and of Closed Graph
Conjugate Pairs Let f (x) is a polynomial function that has real
Positive Semidefinite matrix
§3-3 realization for multivariable systems
GROUPS & THEIR REPRESENTATIONS: a card shuffling approach
Lesson 2.5 The Fundamental Theorem of Algebra
§1-3 Solution of a Dynamical Equation
§1-2 State-Space Description
Gelfand Pairs A. Aizenbud and D. Gourevitch the non compact case
Preliminary.
I. Finite Field Algebra.
Representation Theory
Lecture 43 Section 10.1 Wed, Apr 6, 2005
Theorems about LINEAR MAPPINGS.
Chapter 3 Canonical Form and Irreducible Realization of Linear Time-invariant Systems.
Using Conjugate Symmetries to Enhance Simulation Performance
MA5242 Wavelets Lecture 1 Numbers and Vector Spaces
THE DIMENSION OF A VECTOR SPACE
Preliminaries on normed vector space
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