Decay of matrix coefficients of unitary representations of semisimple groups Michael G Cowling June 18, 2013 1 / 20
Grazie It is nice to be in Italy again! 2 / 20
Introduction ◮ Structure of semisimple Lie groups ◮ Representations of semisimple Lie groups ◮ Decay of matrix coefficients of irreducible representations ◮ Better control of the decay of matrix coefficients. 3 / 20
Semisimple Lie groups “Semisimple Lie group” means a connected Lie group G whose Lie algebra g is a sum of simple ideals. Examples: SL( n , R ), SL( n , C ), SO( p , q ), SU( p , q ), Sp( p , q ), E 8 . 4 / 20
Semisimple Lie groups “Semisimple Lie group” means a connected Lie group G whose Lie algebra g is a sum of simple ideals. Examples: SL( n , R ), SL( n , C ), SO( p , q ), SU( p , q ), Sp( p , q ), E 8 . Every such G has ◮ a maximal compact subgroup K , ◮ a maximal simply connected abelian subgroup A , ◮ a Cartan decomposition G = KA + K , where A + is a cone in A . 4 / 20
Semisimple Lie groups “Semisimple Lie group” means a connected Lie group G whose Lie algebra g is a sum of simple ideals. Examples: SL( n , R ), SL( n , C ), SO( p , q ), SU( p , q ), Sp( p , q ), E 8 . Every such G has ◮ a maximal compact subgroup K , ◮ a maximal simply connected abelian subgroup A , ◮ a Cartan decomposition G = KA + K , where A + is a cone in A . The Lie algebra a is a vector space with a canonical inner product, and we can identify a and a ∗ . But usually we distinguish them. An element λ of a ∗ or a ∗ C gives a homomorphism a �→ exp( λ log a ) from A to C . We often write exp H �→ exp( λ H ) instead. 4 / 20
Roots We may write � g = g 0 ⊕ g α , α ∈ Σ where [ H , X ] = α ( H ) X for all H ∈ a and all X ∈ g α . The hyper- planes { H ∈ a : α ( H ) = 0 } , where α ∈ Σ, divide a into cones. We call one of these the positive cone, a + . ∀ H ∈ a + . We order a ∗ : β ≤ γ ⇐ ⇒ β ( H ) ≤ γ ( H ) Let ρ = 1 α ∈ Σ + dim( g α ) α ; then ρ ∈ ( a ∗ ) + , the cone in a ∗ � 2 corresponding to a + under the identification of a and a ∗ . 5 / 20
Haar measure on G � � � � f ( k exp( H ) k ′ ) w ( H ) dk dH dk ′ , f ( x ) dx = a + G K K where w ( H ) = � j exp( β j H ): the dominant term is exp(2 ρ H ). 6 / 20
Haar measure on G � � � � f ( k exp( H ) k ′ ) w ( H ) dk dH dk ′ , f ( x ) dx = a + G K K where w ( H ) = � j exp( β j H ): the dominant term is exp(2 ρ H ). Suppose that β ∈ a ∗ . Define B β : G → R + by B β ( k exp( H ) k ′ ) = (1 + | H | ) N exp(( β − ρ ) H ) for all k , k ′ ∈ K and all H ∈ a + , where N ∈ N depends on G . 6 / 20
Haar measure on G � � � � f ( k exp( H ) k ′ ) w ( H ) dk dH dk ′ , f ( x ) dx = a + G K K where w ( H ) = � j exp( β j H ): the dominant term is exp(2 ρ H ). Suppose that β ∈ a ∗ . Define B β : G → R + by B β ( k exp( H ) k ′ ) = (1 + | H | ) N exp(( β − ρ ) H ) for all k , k ′ ∈ K and all H ∈ a + , where N ∈ N depends on G . Then � a + (1 + | H | ) Nq exp( q ( β − ρ ) H ) exp(2 ρ H ) dH < ∞ � B β � q q = if and only if q ( β − ρ ) < − 2 ρ , that is, if and only if q > q 0 , say. 6 / 20
Haar measure on G � � � � f ( k exp( H ) k ′ ) w ( H ) dk dH dk ′ , f ( x ) dx = a + G K K where w ( H ) = � j exp( β j H ): the dominant term is exp(2 ρ H ). Suppose that β ∈ a ∗ . Define B β : G → R + by B β ( k exp( H ) k ′ ) = (1 + | H | ) N exp(( β − ρ ) H ) for all k , k ′ ∈ K and all H ∈ a + , where N ∈ N depends on G . Then � a + (1 + | H | ) Nq exp( q ( β − ρ ) H ) exp(2 ρ H ) dH < ∞ � B β � q q = if and only if q ( β − ρ ) < − 2 ρ , that is, if and only if q > q 0 , say. Write f ∈ L q + ( G ) if f ∈ L q + ε ( G ) for all ε ∈ R + . 6 / 20
Unitary representations and B ( G ) Let ¯ G denote the “set” of “all” continuous unitary representations π of G on Hilbert spaces H π . 7 / 20
Unitary representations and B ( G ) Let ¯ G denote the “set” of “all” continuous unitary representations π of G on Hilbert spaces H π . A matrix entry is a function u of the form � πξ, η � , that is, u ( x ) = � π ( x ) ξ, η � ∀ x ∈ G , where π ∈ ¯ G and ξ, η ∈ H π . 7 / 20
Unitary representations and B ( G ) Let ¯ G denote the “set” of “all” continuous unitary representations π of G on Hilbert spaces H π . A matrix entry is a function u of the form � πξ, η � , that is, u ( x ) = � π ( x ) ξ, η � ∀ x ∈ G , where π ∈ ¯ G and ξ, η ∈ H π . Next B ( G ) = { u ∈ C ( G ) : u = � πξ, η � , π ∈ ¯ G , ξ, η ∈ H π } ; the same function u may arise in different ways. 7 / 20
Unitary representations and B ( G ) Let ¯ G denote the “set” of “all” continuous unitary representations π of G on Hilbert spaces H π . A matrix entry is a function u of the form � πξ, η � , that is, u ( x ) = � π ( x ) ξ, η � ∀ x ∈ G , where π ∈ ¯ G and ξ, η ∈ H π . Next B ( G ) = { u ∈ C ( G ) : u = � πξ, η � , π ∈ ¯ G , ξ, η ∈ H π } ; the same function u may arise in different ways. For u ∈ B ( G ), � u � B = inf {� ξ � � η � : u = � πξ, η � , π ∈ ¯ G , ξ, η ∈ H π } . 7 / 20
Unitary representations and B ( G ) Let ¯ G denote the “set” of “all” continuous unitary representations π of G on Hilbert spaces H π . A matrix entry is a function u of the form � πξ, η � , that is, u ( x ) = � π ( x ) ξ, η � ∀ x ∈ G , where π ∈ ¯ G and ξ, η ∈ H π . Next B ( G ) = { u ∈ C ( G ) : u = � πξ, η � , π ∈ ¯ G , ξ, η ∈ H π } ; the same function u may arise in different ways. For u ∈ B ( G ), � u � B = inf {� ξ � � η � : u = � πξ, η � , π ∈ ¯ G , ξ, η ∈ H π } . With pointwise operations, B ( G ) is a Banach algebra. 7 / 20
Restricting unitary representations to K If π ∈ ¯ � G , then π K = � K n τ τ , and H π = � K n τ H τ . τ ∈ ˆ τ ∈ ˆ � Let P τ be the orthogonal projection of H π onto n τ H τ . We say that ξ ∈ H π is τ -isotypic if P τ ξ = ξ , and K -finite if it is a finite linear combination of isotypic vectors. 8 / 20
Restricting unitary representations to K If π ∈ ¯ � G , then π K = � K n τ τ , and H π = � K n τ H τ . τ ∈ ˆ τ ∈ ˆ � Let P τ be the orthogonal projection of H π onto n τ H τ . We say that ξ ∈ H π is τ -isotypic if P τ ξ = ξ , and K -finite if it is a finite linear combination of isotypic vectors. As usual, we write ˆ G for the subset of ¯ G consisting of irreducible representations. 8 / 20
Restricting unitary representations to K If π ∈ ¯ � G , then π K = � K n τ τ , and H π = � K n τ H τ . τ ∈ ˆ τ ∈ ˆ � Let P τ be the orthogonal projection of H π onto n τ H τ . We say that ξ ∈ H π is τ -isotypic if P τ ξ = ξ , and K -finite if it is a finite linear combination of isotypic vectors. As usual, we write ˆ G for the subset of ¯ G consisting of irreducible representations. Theorem For all π ∈ ˆ G and all τ ∈ ˆ K, n τ ≤ dim( H τ ) . 8 / 20
Restricting attention to A Let π ∈ ¯ G , σ, τ ∈ ˆ K . Define Φ in C ( A + , Hom( σ, τ )) by Φ( a ) = P τ π ( a ) P σ ∀ a ∈ A . Note that Φ depends on π , σ , and τ . 9 / 20
Restricting attention to A Let π ∈ ¯ G , σ, τ ∈ ˆ K . Define Φ in C ( A + , Hom( σ, τ )) by Φ( a ) = P τ π ( a ) P σ ∀ a ∈ A . Note that Φ depends on π , σ , and τ . If ξ is σ -isotypic and η is τ -isotypic, then � π ( kak ′ ) ξ, η � � π ( a ) π ( k ′ ) ξ, π ( k ) ∗ η � = Φ( a ) π ( k ′ ) ξ, π ( k − 1 ) η � � = for all k , k ′ ∈ K and all a ∈ A . Thus the matrix-valued functions Φ encapsulate the behaviour of π . 9 / 20
Restricting attention to A Let π ∈ ¯ G , σ, τ ∈ ˆ K . Define Φ in C ( A + , Hom( σ, τ )) by Φ( a ) = P τ π ( a ) P σ ∀ a ∈ A . Note that Φ depends on π , σ , and τ . If ξ is σ -isotypic and η is τ -isotypic, then � π ( kak ′ ) ξ, η � � π ( a ) π ( k ′ ) ξ, π ( k ) ∗ η � = Φ( a ) π ( k ′ ) ξ, π ( k − 1 ) η � � = for all k , k ′ ∈ K and all a ∈ A . Thus the matrix-valued functions Φ encapsulate the behaviour of π . If π ∈ ˆ G , then Φ( a ) is finite-dimensional. 9 / 20
Asymptotic behaviour of matrix coefficients Theorem (Harish-Chandra) Suppose that π ∈ ˆ G, and σ, τ ∈ ˆ K. Then Φ = � j Φ j , and for each j there exist α j ∈ a ∗ C and a polynomial p j , independent of σ and τ , and ϕ j ∈ Hom( σ, τ ) such that as H → ∞ ∈ a + . Φ j (exp( H )) ≍ p j ( H ) exp(( α j − ρ ) H ) ϕ j The indices j may be chosen such that Re α 1 ≥ Re α j when j � = 1 , and deg p j ≤ N; the integer N depends only on G. The number of terms in the sum is bounded by a quantity depending on G. 10 / 20
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