Howes correspondence and characters for dual pairs over Archimedean - PowerPoint PPT Presentation

Howe’s correspondence and characters for dual pairs over Archimedean and non-Archimedean fields Tomasz Przebinda University of Oklahoma Norman, OK, USA Symmetries in Geometry, Analysis and Spectral Theory, Paderborn, July 23-27, 2018 On the occasion of Joachim Hilgert’s 60th Birthday Tomasz Przebinda (University of Oklahoma) Howe’s correspondence and characters 1 / 20

The Cauchy determinantal identity, 1812 � | h k 1 h k 2 ... h k n | | h ′ k 1 h ′ k 2 ... h ′ k n | 1 � j ) = | h n − 1 h n − 2 ... h 0 | · 1 ≤ i , j ≤ n ( 1 − h i h ′ | h ′ n − 1 h ′ n − 2 ... h ′ 0 | k 1 > k 2 >...> k n   h k 1 1 h k 2 1 ... h k n 1   h k 1 2 h k 2 2 .. . h k n   | h k 1 h k 2 ... h k n where n | = det  2   ..................  h k 1 n h k 2 n ... h k n n Tomasz Przebinda (University of Oklahoma) Howe’s correspondence and characters 2 / 20

An interpretation of Cauchy’s identity The formula ω ( g , g ′ ) x = gxg ′ t ( x ∈ M n , n ( C ) , ( g , g ′ ) ∈ U n × U n ) defines a representation ω of the group U n × U n on space H ω = Sym ( M n , n ( C )) of the symmetric tensors of M n , n ( C ) . Taking the trace of of ω ( g , g ′ ) , one obtains the character formula � (Π = Π ′ ∈ � Θ ω ( g , g ′ ) = Θ Π ( g )Θ Π ′ ( g ′ ) U n ) . Π Hence one deduces the decomposition � H ω = H Π ⊗ H Π ′ . Π We get a correspondence of representations Π ↔ Π ′ and a character formula � Θ ω ( g , g ′ )Θ Π ′ ( g ′− 1 ) dg . Θ Π ( g ) = U n Tomasz Przebinda (University of Oklahoma) Howe’s correspondence and characters 3 / 20

Gaussians and Weil factors on a field F = R or a p -adic field (finite commutative extension of Q p ), p � = 2; dx the Haar measure on F normalized so that the volume of the closed unit ball is 1. If F = R , then choose χ ( r ) = e 2 π ir , r ∈ R , and define � χ ( 1 2 ( a + ib ) x 2 ) dx , γ ( a ) = lim b → 0 + R | a | − 1 π i 4 sgn ( a ) 2 γ W ( a ) , ( a ∈ R \ { 0 } ) . = γ W ( a ) = e If F � = R , then choose a unitary character χ : F → C × of the additive group F , and define � χ ( 1 2 ( a ) x 2 ) dx , γ ( a ) = lim r →∞ x ∈ F , | x | < r 2 γ W ( a ) , γ W ( a ) 8 = 1 | a | − 1 ( a ∈ F \ { 0 } ) . = γ W is the Weil factor. Tomasz Przebinda (University of Oklahoma) Howe’s correspondence and characters 4 / 20

Gaussians and Weil factors on a vector space U finite dimensional vector space over F with Haar measure µ U ; q a nondegenerate quadratic form on U. If F = R , then define � χ ( 1 γ ( q ) = 2 ( q + ip )( u )) d µ U ( u ) , lim p → 0 U | γ ( q ) | = χ ( 1 γ ( q ) γ W ( q ) = 4 sgn ( q )) . If F � = R , then define � χ ( 1 γ ( q ) = 2 q ( u )) d µ U ( u ) , lim r →∞ u ∈ U , | u | < r γ ( q ) | γ ( q ) | , γ W ( a ) 8 = 1 . γ W ( q ) = Tomasz Przebinda (University of Oklahoma) Howe’s correspondence and characters 5 / 20

Determinants ( W , �· , ·� ) ; Sp ∋ g . If F = R , pick J ∈ sp , J 2 = − I , B ( · . · ) = � J · , ·� > 0. Define det ( g − 1 : W / Ker ( g − 1 ) → ( g − 1 ) W ) = det ( � ( g − 1 ) w i , w j � 1 ≤ i , j ≤ m ) , where w 1 , . . . , w m is any B -orthonormal basis of Ker ( g − 1 ) ⊥ B ⊆ W. If F � = R , fix a lattice L ⊆ W and the corresponding norm N L ( w ) = inf {| a | − 1 : a ∈ F × , aw ∈ L} ( w ∈ W ) . Let o F ⊆ F denote the ring of integers. Define det ( g − 1 : W / Ker ( g − 1 ) → ( g − 1 ) W ) F ) 2 ∈ F × / ( o × F ) 2 , = det ( � ( g − 1 ) w i , w j � 1 ≤ i , j ≤ m )( o × where w 1 , . . . , w m are such that the spaces F w 1 , . . . , F w m , Ker ( g − 1 ) span W and are N L -orthogonal, i.e. N L ( a 1 w 1 + · · · + a m w m + w ) = max { N L ( a 1 w 1 ) , . . . , N L ( a m w m ) , N L ( w ) } . Tomasz Przebinda (University of Oklahoma) Howe’s correspondence and characters 6 / 20

The Metaplectic Group [A.-M. Aubert and T.P ., 2014] For g , g 1 , g 2 ∈ Sp , let Θ 2 ( g ) = γ ( 1 ) 2 dim ( g − 1 ) W − 2 � � 2 γ ( det ( g − 1 : W / Ker ( g − 1 ) → ( g − 1 ) W )) �� � � � Θ 2 ( g 1 g 2 ) � � C ( g 1 , g 2 ) = � γ W ( q g 1 , g 2 ) , � Θ 2 ( g 1 )Θ 2 ( g 2 ) where q g 1 , g 2 ( u ′ , u ′′ ) = 1 2 � ( g 1 + 1 )( g 1 − 1 ) − 1 u ′ , u ′′ � + 1 2 � ( g 2 + 1 )( g 2 − 1 ) − 1 u ′ , u ′′ � ( u ′ , u ′′ ∈ ( g 1 − 1 ) W ∩ ( g 2 − 1 ) W ) . The Metaplectic Group � � g = ( g , ξ ) ∈ Sp × C , ξ 2 = Θ 2 ( g ) � ˜ Sp = ( g 1 , ξ 1 )( g 2 , ξ 2 ) = ( g 1 g 2 , ξ 1 ξ 2 C ( g 1 , g 2 )) . Tomasz Przebinda (University of Oklahoma) Howe’s correspondence and characters 7 / 20

Normalization of Haar measures on vector spaces Let F = R . For any subspace U ⊆ W we normalize the Haar measure µ U on U so that the volume of the unit cube with respect to form B is 1. If V ⊆ U, then B induces a positive definite form on the quotient U / V and hence a normalized Haar measure µ U / V so that the volume of the unit cube is 1. Let F � = R . For any subspace U ⊆ W we normalize the Haar measure µ U on U so that the volume of the lattice L ∩ U is 1. If V ⊆ U, then we normalized Haar measure µ U / V so that the volume of the lattice ( L ∩ U + V ) / V is 1. Tomasz Przebinda (University of Oklahoma) Howe’s correspondence and characters 8 / 20

The Weil Representation W = X ⊕ Y a complete polarization. Op : S ∗ ( X × X ) → Hom ( S ( X ) , S ∗ ( X )) � K ( x , x ′ ) v ( x ′ ) d µ X ( x ′ ) . Op ( K ) v ( x ) = X Weyl transform K : S ∗ ( W ) → S ∗ ( X × X ) � � 1 � f ( x − x ′ + y ) χ K ( f )( x , x ′ ) = 2 � y , x + x ′ � d µ Y ( y ) . Y An imaginary Gaussian on ( g − 1 ) W � 1 � 4 � ( g + 1 )( g − 1 ) − 1 χ c ( g ) ( u ) = χ u , u � ( u = ( g − 1 ) w , w ∈ W ) . � �� � c ( g ) g = ( g , ξ ) ∈ � For ˜ Sp define Θ(˜ T (˜ g ) = Θ(˜ ω (˜ g ) = Op ◦ K ◦ T (˜ g ) = ξ, g ) χ c ( g ) µ ( g − 1 ) W , g ) . ( ω, L 2 ( X )) is the Weil representation of � Sp attached to the character χ . Tomasz Przebinda (University of Oklahoma) Howe’s correspondence and characters 9 / 20

Dual Pairs Subgroups G , G ′ ⊆ Sp ( W ) acting reductively on W. G ′ is the centralizer of G in Sp and G is the centralizer of G ′ in Sp . G ′ ⊆ � The preimages � G , � Sp ( W ) are also mutual centralizers in the metaplectic group. For F = R : G , G ′ stable range GL n ( D ) , GL m ( D ) n ≥ 2 m O p , q , Sp 2 n ( R ) p , q ≥ 2 n Sp 2 n ( R ) , O p , q n ≥ p + q O p ( C ) , Sp 2 n ( C ) p ≥ 4 n Sp 2 n ( C ) , O p ( C ) n ≥ p U p , q , U r , s p , q ≥ r + s Sp p , q , O ∗ p , q ≥ n 2 n O ∗ 2 n , Sp p , q n ≥ 2 ( p + q ) Tomasz Przebinda (University of Oklahoma) Howe’s correspondence and characters 10 / 20

Howe’s Correspondence [Howe, Waldspurger, Gan, Gan-Sun] R ( � G ) equivalence classes of irreducible admissible representations. R ( � G , ω ) ⊆ R ( � G ) representations realized as quotients of S ( X ) by closed � G -invariant subspaces. For Π ∈ R ( � G , ω ) let N Π ⊆ S ( X ) be the intersection of all the closed � G -invariant subspaces N ⊆ S ( X ) such that Π is equivalent to S ( X ) / N . Then S ( X ) / N Π is a representation of both � G and � G ′ . It is equivalent to Π ⊗ Π ′ 1 , G ′ has a unique 1 of � 1 of � for some representation Π ′ G ′ . The representation Π ′ irreducible quotient Π ′ ∈ R ( � G ′ , ω ) . Conversely, starting with Π ′ ∈ R ( � G ′ , ω ) and applying the above procedure with the roles of G and G ′ reversed, we arrive at the representation Π ∈ R ( � G , ω ) . The resulting bijection → Π ′ ∈ R ( � R ( � G ′ , ω ) G , ω ) ∋ Π ← is called Howe’s correspondence, or local θ correspondence, for the pair G , G ′ . Tomasz Przebinda (University of Oklahoma) Howe’s correspondence and characters 11 / 20

The wave front set of a distribution for F = R Let V be a finite dimensional vector space over R . Recall the Fourier transform � c ( V ) , v ∗ ∈ V ∗ ) . F ( φ )( v ∗ ) = φ ( v ) χ ( − v ∗ ( v )) d µ V ( v ) ( φ ∈ C ∞ V The wave front set of a distribution u on V at a point v ∈ V , denoted WF v ( u ) , is the complement of the set of all pairs ( v , v ∗ ) , v ∗ ∈ V ∗ , for which there is a test function φ ∈ C ∞ c ( V ) with φ ( v ) � = 0 and an open cone Γ ⊆ V ∗ containing v ∗ such that |F ( φ u )( v ∗ 1 ) | ≤ C N ( 1 + | v ∗ 1 | ) − N ( v ∗ 1 ∈ Γ , N = 0 , 1 , 2 , ... ) . This notion behaves well under diffeomorphisms. So for any distribution u on a manifold M , one may define WF ( u ) ⊆ T ∗ M as the union of the wave front sets at the individual points. Tomasz Przebinda (University of Oklahoma) Howe’s correspondence and characters 12 / 20