The Chinta-Gunnells action and sums over highest weight crystals - PowerPoint PPT Presentation

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Tokuyama’s Theorem � � G ( b ) · x wt ( b ) . ( x j − v · x i ) · s λ ( x ) = 1 ≤ i < j < r +1 b ∈B λ + ρ Crystal B λ + ρ Schur function s λ ( x ) Sum over the group S r +1 : Position of b in B λ + ρ gives G ( b ) � ∆ v sgn ( w ) · w ( x λ + ρ ) ∆ · w ∈ S r +1 Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Tokuyama’s Theorem � � G ( b ) · x wt ( b ) . ( x j − v · x i ) · s λ ( x ) = 1 ≤ i < j < r +1 b ∈B λ + ρ Crystal B λ + ρ Schur function s λ ( x ) Sum over the group S r +1 : Position of b in B λ + ρ gives G ( b ) � ∆ v sgn ( w ) · w ( x λ + ρ ) ∆ · w ∈ S r +1 Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Tokuyama’s Theorem � � G ( b ) · x wt ( b ) . ( x j − v · x i ) · s λ ( x ) = 1 ≤ i < j < r +1 b ∈B λ + ρ Crystal B λ + ρ Schur function s λ ( x ) Sum over the group S r +1 : Position of b in B λ + ρ gives G ( b ) � ∆ v sgn ( w ) · w ( x λ + ρ ) ∆ · w ∈ S r +1 Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Tokuyama’s Theorem � � G ( b ) · x wt ( b ) . ( x j − v · x i ) · s λ ( x ) = 1 ≤ i < j < r +1 b ∈B λ + ρ Crystal B λ + ρ Schur function s λ ( x ) Sum over the group S r +1 : Position of b in B λ + ρ gives G ( b ) � ∆ v sgn ( w ) · w ( x λ + ρ ) ∆ · w ∈ S r +1 Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Metaplectic Casselman-Shalika � x λ + ρ � � � ∆ v · sgn ( w ) · w � ∆ 1 w ∈ S r +1 The action of W on C (Λ) can be modified to depend on n . (Chinta-Gunnells, Chinta-Offen, McNamara) � G ( b ) · x wt ( b ) b ∈B λ + ρ The definition of G ( b ) can be modified to involve Gauss-sums (modulo n ). (Brubaker, Bump, Friedberg, McNamara, Zhang) Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Metaplectic Casselman-Shalika � x λ + ρ � � � ∆ v · sgn ( w ) · w � ∆ 1 w ∈ S r +1 The action of W on C (Λ) can be modified to depend on n . (Chinta-Gunnells, Chinta-Offen, McNamara) � G ( b ) · x wt ( b ) b ∈B λ + ρ The definition of G ( b ) can be modified to involve Gauss-sums (modulo n ). (Brubaker, Bump, Friedberg, McNamara, Zhang) Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Metaplectic Casselman-Shalika � x λ + ρ � � � ∆ v · sgn ( w ) · w � ∆ 1 w ∈ S r +1 The action of W on C (Λ) can be modified to depend on n . (Chinta-Gunnells, Chinta-Offen, McNamara) � G ( b ) · x wt ( b ) b ∈B λ + ρ The definition of G ( b ) can be modified to involve Gauss-sums (modulo n ). (Brubaker, Bump, Friedberg, McNamara, Zhang) Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Metaplectic Casselman-Shalika � x λ + ρ � � � ∆ v · sgn ( w ) · w � ∆ 1 w ∈ S r +1 The action of W on C (Λ) can be modified to depend on n . (Chinta-Gunnells, Chinta-Offen, McNamara) � G ( b ) · x wt ( b ) b ∈B λ + ρ The definition of G ( b ) can be modified to involve Gauss-sums (modulo n ). (Brubaker, Bump, Friedberg, McNamara, Zhang) Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Metaplectic Casselman-Shalika � x λ + ρ � � � ∆ v · sgn ( w ) · w � ∆ 1 w ∈ S r +1 The action of W on C (Λ) can be modified to depend on n . (Chinta-Gunnells, Chinta-Offen, McNamara) � G ( b ) · x wt ( b ) b ∈B λ + ρ The definition of G ( b ) can be modified to involve Gauss-sums (modulo n ). (Brubaker, Bump, Friedberg, McNamara, Zhang) Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Sum over Weyl group Sum over crystal ? � � e λ + ρ � � � G ( b ) · e wt ( b ) ∆ v sgn ( w ) w � b ∈B λ + ρ ∆ 1 w ∈ W ? W w , λ ∨ ≈ T w x λ ∨ Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Sum over Weyl group Sum over crystal ? � � e λ + ρ � � � G ( b ) · e wt ( b ) ∆ v sgn ( w ) w � b ∈B λ + ρ ∆ 1 w ∈ W ? W w , λ ∨ ≈ T w x λ ∨ Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Sum over Weyl group Sum over crystal ? � � e λ + ρ � � � G ( b ) · e wt ( b ) ∆ v sgn ( w ) w � b ∈B λ + ρ ∆ 1 w ∈ W Iwahori-Whittaker W w , λ ∨ ≈ T w x λ ∨ Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Iwahori-Whittaker functions and the T w Brubaker-Bump-Licata: express the value of a Whittaker functional on Iwahori-fixed vectors of a principal series representation as T w x λ ∨ relate identities of W w , λ ∨ ≈ T w x λ ∨ to combinatorics of Bott-Samelson resolutions, non-symmetric Macdonald polynomials exploit uniqueness of the Whittaker functional Patnaik: gives a proof of W w , λ ∨ ≈ T w x λ ∨ without exploiting uniqueness the method generalizes to the affine Kac-Moody setting Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Iwahori-Whittaker functions and the T w Brubaker-Bump-Licata: express the value of a Whittaker functional on Iwahori-fixed vectors of a principal series representation as T w x λ ∨ relate identities of W w , λ ∨ ≈ T w x λ ∨ to combinatorics of Bott-Samelson resolutions, non-symmetric Macdonald polynomials exploit uniqueness of the Whittaker functional Patnaik: gives a proof of W w , λ ∨ ≈ T w x λ ∨ without exploiting uniqueness the method generalizes to the affine Kac-Moody setting Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Iwahori-Whittaker functions and the T w Brubaker-Bump-Licata: express the value of a Whittaker functional on Iwahori-fixed vectors of a principal series representation as T w x λ ∨ relate identities of W w , λ ∨ ≈ T w x λ ∨ to combinatorics of Bott-Samelson resolutions, non-symmetric Macdonald polynomials exploit uniqueness of the Whittaker functional Patnaik: gives a proof of W w , λ ∨ ≈ T w x λ ∨ without exploiting uniqueness the method generalizes to the affine Kac-Moody setting Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Iwahori-Whittaker functions and the T w Brubaker-Bump-Licata: express the value of a Whittaker functional on Iwahori-fixed vectors of a principal series representation as T w x λ ∨ relate identities of W w , λ ∨ ≈ T w x λ ∨ to combinatorics of Bott-Samelson resolutions, non-symmetric Macdonald polynomials exploit uniqueness of the Whittaker functional Patnaik: gives a proof of W w , λ ∨ ≈ T w x λ ∨ without exploiting uniqueness the method generalizes to the affine Kac-Moody setting Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Iwahori-Whittaker functions and the T w Brubaker-Bump-Licata: express the value of a Whittaker functional on Iwahori-fixed vectors of a principal series representation as T w x λ ∨ relate identities of W w , λ ∨ ≈ T w x λ ∨ to combinatorics of Bott-Samelson resolutions, non-symmetric Macdonald polynomials exploit uniqueness of the Whittaker functional Patnaik: gives a proof of W w , λ ∨ ≈ T w x λ ∨ without exploiting uniqueness the method generalizes to the affine Kac-Moody setting Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Iwahori-Whittaker functions and the T w Brubaker-Bump-Licata: express the value of a Whittaker functional on Iwahori-fixed vectors of a principal series representation as T w x λ ∨ relate identities of W w , λ ∨ ≈ T w x λ ∨ to combinatorics of Bott-Samelson resolutions, non-symmetric Macdonald polynomials exploit uniqueness of the Whittaker functional Patnaik: gives a proof of W w , λ ∨ ≈ T w x λ ∨ without exploiting uniqueness the method generalizes to the affine Kac-Moody setting Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Iwahori-Whittaker functions and the T w Brubaker-Bump-Licata: express the value of a Whittaker functional on Iwahori-fixed vectors of a principal series representation as T w x λ ∨ relate identities of W w , λ ∨ ≈ T w x λ ∨ to combinatorics of Bott-Samelson resolutions, non-symmetric Macdonald polynomials exploit uniqueness of the Whittaker functional Patnaik: gives a proof of W w , λ ∨ ≈ T w x λ ∨ without exploiting uniqueness the method generalizes to the affine Kac-Moody setting Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Iwahori-Whittaker functions and the T w Brubaker-Bump-Licata: express the value of a Whittaker functional on Iwahori-fixed vectors of a principal series representation as T w x λ ∨ relate identities of W w , λ ∨ ≈ T w x λ ∨ to combinatorics of Bott-Samelson resolutions, non-symmetric Macdonald polynomials exploit uniqueness of the Whittaker functional Patnaik: gives a proof of W w , λ ∨ ≈ T w x λ ∨ without exploiting uniqueness the method generalizes to the affine Kac-Moody setting Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Metaplectic Demazure operators Demazure operators: σ i simple reflection, f ∈ C (Λ): D σ i ( f ) = f − x − n( α ∨ i ) α ∨ i σ i ( f ) 1 − x − n( α ∨ i ) α ∨ i Demazure-Lusztig operators: T σ i ( f ) = (1 − v · x − n( α ∨ i ) α ∨ i ) · D σ i ( f ) − f n n( α ∨ ) = gcd( n , || α ∨ || 2 ) and σ i ( f ) is the Chinta-Gunnells action D σ i , T σ i satisfy Braid-relations − → D w , T w for every w ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Metaplectic Demazure operators Demazure operators: σ i simple reflection, f ∈ C (Λ): D σ i ( f ) = f − x − n( α ∨ i ) α ∨ i σ i ( f ) 1 − x − n( α ∨ i ) α ∨ i Demazure-Lusztig operators: T σ i ( f ) = (1 − v · x − n( α ∨ i ) α ∨ i ) · D σ i ( f ) − f n n( α ∨ ) = gcd( n , || α ∨ || 2 ) and σ i ( f ) is the Chinta-Gunnells action D σ i , T σ i satisfy Braid-relations − → D w , T w for every w ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Metaplectic Demazure operators Demazure operators: σ i simple reflection, f ∈ C (Λ): D σ i ( f ) = f − x − n( α ∨ i ) α ∨ i σ i ( f ) 1 − x − n( α ∨ i ) α ∨ i Demazure-Lusztig operators: T σ i ( f ) = (1 − v · x − n( α ∨ i ) α ∨ i ) · D σ i ( f ) − f n n( α ∨ ) = gcd( n , || α ∨ || 2 ) and σ i ( f ) is the Chinta-Gunnells action D σ i , T σ i satisfy Braid-relations − → D w , T w for every w ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Metaplectic Demazure operators Demazure operators: σ i simple reflection, f ∈ C (Λ): D σ i ( f ) = f − x − n( α ∨ i ) α ∨ i σ i ( f ) 1 − x − n( α ∨ i ) α ∨ i Demazure-Lusztig operators: T σ i ( f ) = (1 − v · x − n( α ∨ i ) α ∨ i ) · D σ i ( f ) − f n n( α ∨ ) = gcd( n , || α ∨ || 2 ) and σ i ( f ) is the Chinta-Gunnells action D σ i , T σ i satisfy Braid-relations − → D w , T w for every w ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Metaplectic Demazure operators Demazure operators: σ i simple reflection, f ∈ C (Λ): D σ i ( f ) = f − x − n( α ∨ i ) α ∨ i σ i ( f ) 1 − x − n( α ∨ i ) α ∨ i Demazure-Lusztig operators: T σ i ( f ) = (1 − v · x − n( α ∨ i ) α ∨ i ) · D σ i ( f ) − f n n( α ∨ ) = gcd( n , || α ∨ || 2 ) and σ i ( f ) is the Chinta-Gunnells action D σ i , T σ i satisfy Braid-relations − → D w , T w for every w ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Identities for the long word Theorem (Chinta, Gunnells, P.) � � D w 0 = 1 e n( α ) α · w . · sgn ( w ) · � ∆ w ∈ W α ∈ Φ( w − 1 ) � � ∆ v · D w 0 = T w . w ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Identities for the long word Theorem (Chinta, Gunnells, P.) � � D w 0 = 1 e n( α ) α · w . · sgn ( w ) · � ∆ w ∈ W α ∈ Φ( w − 1 ) � � ∆ v · D w 0 = T w . w ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Identities for the long word Theorem (Chinta, Gunnells, P.) � � D w 0 = 1 e n( α ) α · w . · sgn ( w ) · � ∆ w ∈ W α ∈ Φ( w − 1 ) � � ∆ v · D w 0 = T w . w ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Identities for the long word Theorem (Chinta, Gunnells, P.) � � D w 0 = 1 e n( α ) α · w . · sgn ( w ) · � ∆ w ∈ W α ∈ Φ( w − 1 ) � � ∆ v · D w 0 = T w . w ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Sum over Weyl group Sum over crystal ? � � e λ + ρ � � � G ( b ) · e wt ( b ) ∆ v sgn ( w ) w � b ∈B λ + ρ ∆ 1 w ∈ W Hecke symmetrizer � T w w ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Sum over Weyl group Sum over crystal ? � � e λ + ρ � � � G ( b ) · e wt ( b ) ∆ v sgn ( w ) w � b ∈B λ + ρ ∆ 1 w ∈ W Hecke symmetrizer � T w w ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Sum over Weyl group Sum over crystal ? � � e λ + ρ � � � G ( b ) · e wt ( b ) ∆ v sgn ( w ) w � b ∈B λ + ρ ∆ 1 w ∈ W Hecke symmetrizer � T w w ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Sum over Weyl group Sum over crystal ? � � e λ + ρ � � � G ( b ) · e wt ( b ) ∆ v sgn ( w ) w � b ∈B λ + ρ ∆ 1 w ∈ W � T w = ∆ v D w 0 ∼ ∆ v χ λ w ∈ W Hecke symmetrizer � T w w ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Sum over Weyl group Sum over crystal ? � � e λ + ρ � � � G ( b ) · e wt ( b ) ∆ v sgn ( w ) w � b ∈B λ + ρ ∆ 1 w ∈ W Hecke symmetrizer � T w w ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Sum over Weyl group Sum over crystal ? � � e λ + ρ � � � G ( b ) · e wt ( b ) ∆ v sgn ( w ) w � b ∈B λ + ρ ∆ 1 w ∈ W ? Hecke symmetrizer � T w w ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Theorem (P.)   � �  · x λ ∨ x − ρ G ( b ) x wt ( b ) T u = u ≤ w b ∈B ( w ) λ + ρ Sum over the Weyl B ( w ) λ + ρ Demazure subcrystal group, bounded in the Bruhat order Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Theorem (P.)   � �  · x λ ∨ x − ρ G ( b ) x wt ( b ) T u = u ≤ w b ∈B ( w ) λ + ρ Sum over the Weyl B ( w ) λ + ρ Demazure subcrystal group, bounded in the Bruhat order Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Theorem (P.)   � �  · x λ ∨ x − ρ G ( b ) x wt ( b ) T u = u ≤ w b ∈B ( w ) λ + ρ Sum over the Weyl B ( w ) λ + ρ Demazure subcrystal group, bounded in the Bruhat order Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Theorem (P.)   � �  · x λ ∨ x − ρ G ( b ) x wt ( b ) T u = u ≤ w b ∈B ( w ) λ + ρ Sum over the Weyl B ( w ) λ + ρ Demazure subcrystal group, bounded in the Bruhat order Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Sum over Weyl group Sum over crystal ? � � e λ + ρ � � � G ( b ) · e wt ( b ) ∆ v sgn ( w ) w � b ∈B λ + ρ ∆ 1 w ∈ W Induction by rank Hecke symmetrizer � T w w ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Sum over Weyl group Sum over crystal ? � � e λ + ρ � � � G ( b ) · e wt ( b ) ∆ v sgn ( w ) w � b ∈B λ + ρ ∆ 1 w ∈ W Induction by rank Hecke symmetrizer � T w w ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Sum over Weyl group Sum over crystal ? � � e λ + ρ � � � G ( b ) · e wt ( b ) ∆ v sgn ( w ) w � b ∈B λ + ρ ∆ 1 w ∈ W Induction by rank Hecke symmetrizer � T w w ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Induction by rank · Joint work in progress with Paul E. Gunnells: this technique extends to Cartan type D . Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Induction by rank � � T w = T w · (1 + T r + T r T r − 1 + · · · + T r T r − 1 · · · T 1 ) w ∈ W ( r ) w ∈ W ( r − 1) Joint work in progress with Paul E. Gunnells: this technique extends to Cartan type D . Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Induction by rank � � T w = T w · (1 + T r + T r T r − 1 + · · · + T r T r − 1 · · · T 1 ) w ∈ W ( r ) w ∈ W ( r − 1) Joint work in progress with Paul E. Gunnells: this technique extends to Cartan type D . Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Induction by rank � � T w = T w · (1 + T r + T r T r − 1 + · · · + T r T r − 1 · · · T 1 ) w ∈ W ( r ) w ∈ W ( r − 1) Joint work in progress with Paul E. Gunnells: this technique extends to Cartan type D . Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Induction by rank � � T w = T w · (1 + T r + T r T r − 1 + · · · + T r T r − 1 · · · T 1 ) w ∈ W ( r ) w ∈ W ( r − 1) Joint work in progress with Paul E. Gunnells: this technique extends to Cartan type D . Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Demazure-Lusztig operator T w Questions: interpretation as metaplectic Iwahori-Whittaker function infinite dimensional case: metaplectic Kac-Moody Whittaker functions identities and relationship to Weyl-Kac character � � � e λ ∼ m ∆ v χ λ T w w ∈ W Joint work with Manish Patnaik Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Demazure-Lusztig operator T w Questions: interpretation as metaplectic Iwahori-Whittaker function infinite dimensional case: metaplectic Kac-Moody Whittaker functions identities and relationship to Weyl-Kac character � � � e λ ∼ m ∆ v χ λ T w w ∈ W Joint work with Manish Patnaik Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Demazure-Lusztig operator T w Questions: interpretation as metaplectic Iwahori-Whittaker function infinite dimensional case: metaplectic Kac-Moody Whittaker functions identities and relationship to Weyl-Kac character � � � e λ ∼ m ∆ v χ λ T w w ∈ W Joint work with Manish Patnaik Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Demazure-Lusztig operator T w Questions: interpretation as metaplectic Iwahori-Whittaker function infinite dimensional case: metaplectic Kac-Moody Whittaker functions identities and relationship to Weyl-Kac character � � � e λ ∼ m ∆ v χ λ T w w ∈ W Joint work with Manish Patnaik Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Demazure-Lusztig operator T w Questions: interpretation as metaplectic Iwahori-Whittaker function infinite dimensional case: metaplectic Kac-Moody Whittaker functions identities and relationship to Weyl-Kac character � � � e λ ∼ m ∆ v χ λ T w w ∈ W Joint work with Manish Patnaik Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Demazure-Lusztig operator T w Questions: interpretation as metaplectic Iwahori-Whittaker function infinite dimensional case: metaplectic Kac-Moody Whittaker functions identities and relationship to Weyl-Kac character � � � e λ ∼ m ∆ v χ λ T w w ∈ W Joint work with Manish Patnaik Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Demazure-Lusztig operator T w Questions: interpretation as metaplectic Iwahori-Whittaker function infinite dimensional case: metaplectic Kac-Moody Whittaker functions identities and relationship to Weyl-Kac character � � � e λ ∼ m ∆ v χ λ T w w ∈ W Joint work with Manish Patnaik Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Metaplectic Iwahori-Whittaker functions � W u , λ ∨ � W ( π λ ∨ ) = q − 2 � ρ , λ ∨ � · � � W u , λ ∨ u ∈ W Theorem (Patnaik, P.) Let w , w ′ ∈ W and w = σ i w ′ with ℓ ( w ) = ℓ ( w ′ ) + 1 : T σ i ( � W w ′ , λ ∨ ) = � W w , λ ∨ W w , λ ∨ = q � ρ , λ ∨ � · T w ( e λ ∨ ). Corollary: � (New proof of the metaplectic Casselman-Shalika formula.) Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Metaplectic Iwahori-Whittaker functions � W u , λ ∨ � W ( π λ ∨ ) = q − 2 � ρ , λ ∨ � · � � W u , λ ∨ u ∈ W Theorem (Patnaik, P.) Let w , w ′ ∈ W and w = σ i w ′ with ℓ ( w ) = ℓ ( w ′ ) + 1 : T σ i ( � W w ′ , λ ∨ ) = � W w , λ ∨ W w , λ ∨ = q � ρ , λ ∨ � · T w ( e λ ∨ ). Corollary: � (New proof of the metaplectic Casselman-Shalika formula.) Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Metaplectic Iwahori-Whittaker functions � W u , λ ∨ � W ( π λ ∨ ) = q − 2 � ρ , λ ∨ � · � � W u , λ ∨ u ∈ W Theorem (Patnaik, P.) Let w , w ′ ∈ W and w = σ i w ′ with ℓ ( w ) = ℓ ( w ′ ) + 1 : T σ i ( � W w ′ , λ ∨ ) = � W w , λ ∨ W w , λ ∨ = q � ρ , λ ∨ � · T w ( e λ ∨ ). Corollary: � (New proof of the metaplectic Casselman-Shalika formula.) Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Metaplectic Iwahori-Whittaker functions � W u , λ ∨ � W ( π λ ∨ ) = q − 2 � ρ , λ ∨ � · � � W u , λ ∨ u ∈ W Theorem (Patnaik, P.) Let w , w ′ ∈ W and w = σ i w ′ with ℓ ( w ) = ℓ ( w ′ ) + 1 : T σ i ( � W w ′ , λ ∨ ) = � W w , λ ∨ W w , λ ∨ = q � ρ , λ ∨ � · T w ( e λ ∨ ). Corollary: � (New proof of the metaplectic Casselman-Shalika formula.) Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Metaplectic Iwahori-Whittaker functions � W u , λ ∨ � W ( π λ ∨ ) = q − 2 � ρ , λ ∨ � · � � W u , λ ∨ u ∈ W Theorem (Patnaik, P.) Let w , w ′ ∈ W and w = σ i w ′ with ℓ ( w ) = ℓ ( w ′ ) + 1 : T σ i ( � W w ′ , λ ∨ ) = � W w , λ ∨ W w , λ ∨ = q � ρ , λ ∨ � · T w ( e λ ∨ ). Corollary: � (New proof of the metaplectic Casselman-Shalika formula.) Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Nonmetaplectic, affine result: Theorem (Patnaik) � W ( π λ ∨ ) = q � ρ , λ ∨ � · T u ( e λ ∨ ) = m · q � ρ , λ ∨ � · χ λ ∨ u ∈ W Metaplectic context: What is the metaplectic cover of a Kac-Moody group? Issues with the convergence of � T u ( e λ ∨ ) u ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Nonmetaplectic, affine result: Theorem (Patnaik) � W ( π λ ∨ ) = q � ρ , λ ∨ � · T u ( e λ ∨ ) = m · q � ρ , λ ∨ � · χ λ ∨ u ∈ W Metaplectic context: What is the metaplectic cover of a Kac-Moody group? Issues with the convergence of � T u ( e λ ∨ ) u ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Nonmetaplectic, affine result: Theorem (Patnaik) � W ( π λ ∨ ) = q � ρ , λ ∨ � · T u ( e λ ∨ ) = m · q � ρ , λ ∨ � · χ λ ∨ u ∈ W Metaplectic context: What is the metaplectic cover of a Kac-Moody group? Issues with the convergence of � T u ( e λ ∨ ) u ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Nonmetaplectic, affine result: Theorem (Patnaik) � W ( π λ ∨ ) = q � ρ , λ ∨ � · T u ( e λ ∨ ) = m · q � ρ , λ ∨ � · χ λ ∨ u ∈ W Metaplectic context: What is the metaplectic cover of a Kac-Moody group? Issues with the convergence of � T u ( e λ ∨ ) u ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Nonmetaplectic, affine result: Theorem (Patnaik) � W ( π λ ∨ ) = q � ρ , λ ∨ � · T u ( e λ ∨ ) = m · q � ρ , λ ∨ � · χ λ ∨ u ∈ W Metaplectic context: What is the metaplectic cover of a Kac-Moody group? Issues with the convergence of � T u ( e λ ∨ ) u ∈ W Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Well-definedness and Convergence For any w ∈ W we may expand � T w = A u ( w )[ u ] u ≤ w Summing this over W : � � � T w = A u ( w )[ u ] w ∈ W w ∈ W u ≤ w � For a fixed u ∈ W , why is A u ( w ) well-defined? u ≤ w Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Well-definedness and Convergence For any w ∈ W we may expand � T w = A u ( w )[ u ] u ≤ w Summing this over W : � � � T w = A u ( w )[ u ] w ∈ W w ∈ W u ≤ w � For a fixed u ∈ W , why is A u ( w ) well-defined? u ≤ w Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Well-definedness and Convergence For any w ∈ W we may expand � T w = A u ( w )[ u ] u ≤ w Summing this over W : � � � T w = A u ( w )[ u ] w ∈ W w ∈ W u ≤ w � For a fixed u ∈ W , why is A u ( w ) well-defined? u ≤ w Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Well-definedness and Convergence For any w ∈ W we may expand � T w = A u ( w )[ u ] u ≤ w Summing this over W : � � � T w = A u ( w )[ u ] w ∈ W w ∈ W u ≤ w � For a fixed u ∈ W , why is A u ( w ) well-defined? u ≤ w Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Results - Patnaik, P. Given F , ( · , · ) : F ∗ × F ∗ → A , G , Q : Λ ∨ → Z , B.There exists 1 → A → E → G → 1 such that restricted to the torus H ( λ ∨ , µ ∨ ∈ Λ ∨ , s , t ∈ F ∗ , s λ ∨ , t µ ∨ ∈ H ): [ s λ ∨ , t µ ∨ ] = ( s , t ) B( λ ∨ , µ ∨ ) .   � � W ( π λ ∨ ) = m Φ ∨ � α ∨  w ⋆ e λ ∨ , ( − 1) ℓ ( w )  e − � n ∆ Φ ∨ n w ∈ W α ∨ ∈ Φ ∨ � n ( w ) Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Results - Patnaik, P. Given F , ( · , · ) : F ∗ × F ∗ → A , G , Q : Λ ∨ → Z , B.There exists 1 → A → E → G → 1 such that restricted to the torus H ( λ ∨ , µ ∨ ∈ Λ ∨ , s , t ∈ F ∗ , s λ ∨ , t µ ∨ ∈ H ): [ s λ ∨ , t µ ∨ ] = ( s , t ) B( λ ∨ , µ ∨ ) .   � � W ( π λ ∨ ) = m Φ ∨ � α ∨  w ⋆ e λ ∨ , ( − 1) ℓ ( w )  e − � n ∆ Φ ∨ n w ∈ W α ∨ ∈ Φ ∨ � n ( w ) Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Results - Patnaik, P. Given F , ( · , · ) : F ∗ × F ∗ → A , G , Q : Λ ∨ → Z , B.There exists 1 → A → E → G → 1 such that restricted to the torus H ( λ ∨ , µ ∨ ∈ Λ ∨ , s , t ∈ F ∗ , s λ ∨ , t µ ∨ ∈ H ): [ s λ ∨ , t µ ∨ ] = ( s , t ) B( λ ∨ , µ ∨ ) .   � � W ( π λ ∨ ) = m Φ ∨ � α ∨  w ⋆ e λ ∨ , ( − 1) ℓ ( w )  e − � n ∆ Φ ∨ n w ∈ W α ∨ ∈ Φ ∨ � n ( w ) Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Results - Patnaik, P. Given F , ( · , · ) : F ∗ × F ∗ → A , G , Q : Λ ∨ → Z , B.There exists 1 → A → E → G → 1 such that restricted to the torus H ( λ ∨ , µ ∨ ∈ Λ ∨ , s , t ∈ F ∗ , s λ ∨ , t µ ∨ ∈ H ): [ s λ ∨ , t µ ∨ ] = ( s , t ) B( λ ∨ , µ ∨ ) .   � � W ( π λ ∨ ) = m Φ ∨ � α ∨  w ⋆ e λ ∨ , ( − 1) ℓ ( w )  e − � n ∆ Φ ∨ n w ∈ W α ∨ ∈ Φ ∨ � n ( w ) Anna Pusk´ as University of Massachusetts, Amherst

Motivation Identities of operators Metaplectic Tokuyama Metaplectic Kac-Moody Correction factor Results - Patnaik, P. Given F , ( · , · ) : F ∗ × F ∗ → A , G , Q : Λ ∨ → Z , B.There exists 1 → A → E → G → 1 such that restricted to the torus H ( λ ∨ , µ ∨ ∈ Λ ∨ , s , t ∈ F ∗ , s λ ∨ , t µ ∨ ∈ H ): [ s λ ∨ , t µ ∨ ] = ( s , t ) B( λ ∨ , µ ∨ ) .   � � W ( π λ ∨ ) = m Φ ∨ � α ∨  w ⋆ e λ ∨ , ( − 1) ℓ ( w )  e − � n ∆ Φ ∨ n w ∈ W α ∨ ∈ Φ ∨ � n ( w ) Anna Pusk´ as University of Massachusetts, Amherst