On classification of unitary highest weight modules Representation - PowerPoint PPT Presentation

On classification of unitary highest weight modules Representation Theory XVI – Dubrovnik 2019 Vít Tuček Faculty of Science, University of Zagreb

Supported by the QuantiXLie Centre of Excellence, a project cofinanced by the Croatian Government and European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Programme (KK.01.1.1.01.0004). 1

Joint work with Pavle Pandžić, University of Zagreb Vladimír Souček, Charles University 2

Hermitian symmetric spaces G / Q ≃ G 0 / K where G is complex and Q is a parabolic subgroup with abelian nilradical and Levi part of Q is complexification of the maximal compact subgroup K Cartan decomposition: g = k ⊕ p p = p − ⊕ p + q = k ⊕ p + compact roots Φ c & noncompact roots Φ n ρ = ρ k + ρ n W = W k W k 3

Classification of HSS G G 0 K SL ( p + q , C ) SU ( p , q ) S ( U ( p ) × U ( q )) SO ( p + 2 , C ) SO (2 , p ) S ( O ( p ) × O (2)) SO ∗ (2 n ) SO (2 n , C ) U ( n ) Sp (2 n , C ) Sp ( n , R ) U ( n ) E − 14 E C Spin (10) × SO (2) 6 6 E − 25 E C E 6 × SO (2) 7 7 (Plus covers of these.) 4

Unitarizable highest weight modules g = p − ⊕ k ⊕ p + , q = k ⊕ p + finite-dimensional k -module F λ of highest weight λ generalizde Verma module M ( λ ) = U ( g ) ⊗ U ( q ) F λ its unique maximal submodule J ( λ ) ≤ M ( λ ) and the simple quotient L ( λ ) ≃ M ( λ ) / J ( λ ) Shapovalov form � X · u | v � = � u | σ ( X ) · v � where σ is minus the conjugate with respect to the real form g 0 5

Classification of Unitarizable Highest Weight Modules [EHW83]Thomas Enright, Roger Howe, and Nolan Wallach. “A classification of unitary highest weight modules”. In: Representation theory of reductive groups (Park City, Utah, 1982) . Vol. 40. Progr. Math. Boston, MA: Birkhäuser Boston, 1983, pp. 97–143 [EJ90]Thomas J. Enright and Anthony Joseph. “An intrinsic analysis of unitarizable highest weight modules”. In: Mathematische Annalen 288.1 (Dec. 1990), pp. 571–594 [Jak83] Hans Plesner Jakobsen. “Hermitian symmetric spaces and their unitary highest weight modules”. In: Journal of Functional Analysis 52.3 (July 1, 1983), pp. 385–412 6

Classification of UHW Modules M ( λ ) = U ( g ) ⊗ U ( q ) F λ ≃ U ( p − ) ⊗ C F λ = S ( p − ) ⊗ F λ 0 ⊗ C z 7

Classification of UHW Modules M ( λ ) = U ( g ) ⊗ U ( q ) F λ ≃ U ( p − ) ⊗ C F λ = S ( p − ) ⊗ F λ 0 ⊗ C z β . . . maximal non-compact root any weight λ ∈ h ∗ can be written uniquely as λ = λ 0 + z ζ where � ζ, β ∨ � = 1 ζ ⊥ Φ c , & � λ 0 + ρ, β � = 0 7

Classification of UHW Modules M ( λ ) = U ( g ) ⊗ U ( q ) F λ ≃ U ( p − ) ⊗ C F λ = S ( p − ) ⊗ F λ 0 ⊗ C z β . . . maximal non-compact root any weight λ ∈ h ∗ can be written uniquely as λ = λ 0 + z ζ where � ζ, β ∨ � = 1 ζ ⊥ Φ c , & � λ 0 + ρ, β � = 0 set of z ∈ C for which the simple factor of Verma module L ( λ ) is unitarizable: A ( λ 0 ) B ( λ 0 ) 0 C ( λ 0 ) A ( λ 0 ), B ( λ 0 ) and C ( λ 0 ) are real numbers expressible in terms of certain root systems Q ( λ 0 ) and R ( λ 0 ) associated to λ 0 7

Classification of UHW Modules — continued The level of reduction of a simple module L ( λ ) ≃ M ( λ ) / J ( λ ) is the first natural number k for which J ( λ ) ∩ M ( λ ) k � = 0 , where M ( λ ) k = S k ( p − ) ⊗ F λ 8

Classification of UHW Modules — continued The level of reduction of a simple module L ( λ ) ≃ M ( λ ) / J ( λ ) is the first natural number k for which J ( λ ) ∩ M ( λ ) k � = 0 , where M ( λ ) k = S k ( p − ) ⊗ F λ For λ = λ 0 + ( B ( λ 0 ) − kC ( λ 0 )) ζ we have l ( λ ) = k + 1 A ( λ 0 ) B ( λ 0 ) 0 1 +1 8

Example: G = SU ( p , q ) Using standard coordinates and setting n = p + q , we write λ = ( λ 1 , . . . , λ p | λ p +1 , . . . , λ n ) . 9

Example: G = SU ( p , q ) Using standard coordinates and setting n = p + q , we write λ = ( λ 1 , . . . , λ p | λ p +1 , . . . , λ n ) . In this case β = ǫ 1 − ǫ n and ζ = 1 n ( q , . . . , q | − p , . . . , − p ). 9

Example: G = SU ( p , q ) Using standard coordinates and setting n = p + q , we write λ = ( λ 1 , . . . , λ p | λ p +1 , . . . , λ n ) . In this case β = ǫ 1 − ǫ n and ζ = 1 n ( q , . . . , q | − p , . . . , − p ). If λ 1 = · · · = λ i > λ i +1 ≥ · · · ≥ λ p & λ p +1 ≥ · · · ≥ λ n − j > λ n − j +1 = · · · = λ n , 9

Example: G = SU ( p , q ) Using standard coordinates and setting n = p + q , we write λ = ( λ 1 , . . . , λ p | λ p +1 , . . . , λ n ) . In this case β = ǫ 1 − ǫ n and ζ = 1 n ( q , . . . , q | − p , . . . , − p ). If λ 1 = · · · = λ i > λ i +1 ≥ · · · ≥ λ p & λ p +1 ≥ · · · ≥ λ n − j > λ n − j +1 = · · · = λ n , then Q ( λ 0 ) = R ( λ 0 ) is the root system built on the first i and the last j coordinates. 9

Example: G = SU ( p , q ) Using standard coordinates and setting n = p + q , we write λ = ( λ 1 , . . . , λ p | λ p +1 , . . . , λ n ) . In this case β = ǫ 1 − ǫ n and ζ = 1 n ( q , . . . , q | − p , . . . , − p ). If λ 1 = · · · = λ i > λ i +1 ≥ · · · ≥ λ p & λ p +1 ≥ · · · ≥ λ n − j > λ n − j +1 = · · · = λ n , then Q ( λ 0 ) = R ( λ 0 ) is the root system built on the first i and the last j coordinates. Furthermore, A ( λ 0 ) = max { i , j } , while B ( λ 0 ) = i + j − 1. 9

Our results We organize the classification in a different way, considering the Hasse diagrams of the basic cases (sums of fundamental weights), and the (reduced) translation cones over the basic cases. 10

Our results We organize the classification in a different way, considering the Hasse diagrams of the basic cases (sums of fundamental weights), and the (reduced) translation cones over the basic cases. The possibility of organizing the modules into cones was first noticed by Davidson-Enright-Stanke. 10

Our results We organize the classification in a different way, considering the Hasse diagrams of the basic cases (sums of fundamental weights), and the (reduced) translation cones over the basic cases. The possibility of organizing the modules into cones was first noticed by Davidson-Enright-Stanke. The proof in EHW is based on one instance of Dirac inequality. We use a different version of the Dirac inequality and obtain a simpler and more natural proof. 10

Dirac operator Let C ( p ) be the Clifford algebra of p with respect to the Killing form B . 11

Dirac operator Let C ( p ) be the Clifford algebra of p with respect to the Killing form B . Let b i be any basis of p ; let d i be the dual basis with respect to B . 11

Dirac operator Let C ( p ) be the Clifford algebra of p with respect to the Killing form B . Let b i be any basis of p ; let d i be the dual basis with respect to B . Dirac operator: � D = b i ⊗ d i ∈ U ( g ) ⊗ C ( p ) i 11

Dirac operator Let C ( p ) be the Clifford algebra of p with respect to the Killing form B . Let b i be any basis of p ; let d i be the dual basis with respect to B . Dirac operator: � D = b i ⊗ d i ∈ U ( g ) ⊗ C ( p ) i D is independent of b i and K -invariant. 11

Dirac operator D 2 is the spin Laplacian (Parthasarathy): D 2 = − (Cas g ⊗ 1 + � ρ � 2 ) + (Cas k ∆ + � ρ k � 2 ) . 12

Dirac operator D 2 is the spin Laplacian (Parthasarathy): D 2 = − (Cas g ⊗ 1 + � ρ � 2 ) + (Cas k ∆ + � ρ k � 2 ) . Here Cas g , Cas k ∆ are the Casimir elements of U ( g ), U ( k ∆ ); 12

Dirac operator D 2 is the spin Laplacian (Parthasarathy): D 2 = − (Cas g ⊗ 1 + � ρ � 2 ) + (Cas k ∆ + � ρ k � 2 ) . Here Cas g , Cas k ∆ are the Casimir elements of U ( g ), U ( k ∆ ); k ∆ is the diagonal copy of k in U ( g ) ⊗ C ( p ) defined by k ֒ → g ֒ → U ( g ) and k → so ( p ) ֒ → C ( p ) . 12

Dirac operator � D = b i ⊗ d i ∈ U ( g ) ⊗ C ( p ) i D acts on M ⊗ S , where • M is ( g , K )-module • S is the spin module for C ( p ) ( S = � p + , p + acts by wedging and p − acts by contracting) 13

Parthasarathy’s Dirac inequality If M is unitary, then D is self adjoint wrt an inner product. So D 2 ≥ 0 . 14

Parthasarathy’s Dirac inequality If M is unitary, then D is self adjoint wrt an inner product. So D 2 ≥ 0 . By the formula for D 2 , the inequality becomes explicit on any K -type F τ ⊂ M ⊗ S : � τ + ρ k � 2 ≥ � Λ � 2 , where Λ ∈ h ∗ corresponds to the infinitesimal character of M via the Harish-Chandra isomorphism. (For L ( λ ) we have Λ = λ + ρ. ) 14

Parthasarathy’s Dirac inequality If M is unitary, then D is self adjoint wrt an inner product. So D 2 ≥ 0 . By the formula for D 2 , the inequality becomes explicit on any K -type F τ ⊂ M ⊗ S : � τ + ρ k � 2 ≥ � Λ � 2 , where Λ ∈ h ∗ corresponds to the infinitesimal character of M via the Harish-Chandra isomorphism. (For L ( λ ) we have Λ = λ + ρ. ) Each F τ is contained in some F µ ⊗ F ν ⊂ M ⊗ S . All ν are of the form σρ − ρ k , where σ ∈ W is such that σρ is k -dominant, i.e. σ ∈ W k . 14