A path approach to Kostant modules Mrigendra Singh Kushwaha The - PowerPoint PPT Presentation

A path approach to Kostant modules Mrigendra Singh Kushwaha The Institute of Mathematical Sciences (HBNI) Chennai, India. Joint work with K. N. Raghavan and Sankaran Viswanath Mrigendra Singh Kushwaha A path approach to Kostant modules

Notations g : symmetrizable Kac-Moody algebra. h : Cartan subalgebra. b : Borel subalgebra, containing h . W : Weyl group. Λ : integral weight lattice. Λ + : dominant integral weight lattice. V ( λ ) : irreducible integrable representation with highest weight λ ∈ Λ + . Mrigendra Singh Kushwaha A path approach to Kostant modules

Kostant Modules Fix λ, µ dominant integral weights (throughout talk) and w an ele- ment of the Weyl group. Let v λ be a highest weight vector in V ( λ ). Let v w µ be a non-zero vector in the (one-dimensional) weight space of weight w µ in the irreducible representation V ( µ ). The Kostant module K ( λ, w , µ ) is defined to be the cyclic submodule of the ten- sor product V ( λ ) ⊗ V ( µ ) generated by the element v λ ⊗ v w µ : K ( λ, w , µ ) := U g ( v λ ⊗ v w µ ) where U g denotes the universal enveloping algebra of g . Mrigendra Singh Kushwaha A path approach to Kostant modules

Kostant Modules Fix λ, µ dominant integral weights (throughout talk) and w an ele- ment of the Weyl group. Let v λ be a highest weight vector in V ( λ ). Let v w µ be a non-zero vector in the (one-dimensional) weight space of weight w µ in the irreducible representation V ( µ ). The Kostant module K ( λ, w , µ ) is defined to be the cyclic submodule of the ten- sor product V ( λ ) ⊗ V ( µ ) generated by the element v λ ⊗ v w µ : K ( λ, w , µ ) := U g ( v λ ⊗ v w µ ) where U g denotes the universal enveloping algebra of g . Examples w = 1 : K ( λ, 1 , µ ) ∼ = V ( λ + µ ) . w = w 0 the longest element ( if g is finite dimensional ) : K ( λ, w 0 , µ ) = V ( λ ) ⊗ V ( µ ) . Mrigendra Singh Kushwaha A path approach to Kostant modules

Filtration of V ( λ ) ⊗ V ( µ ) by Kostant modules Observation 1. Let W λ and W µ be the stabilizers of dominant inte- gral weights λ and µ respectively. Then K ( λ, w 1 , µ ) = K ( λ, w 2 , µ ) if W λ w 1 W µ = W λ w 2 W µ for w 1 , w 2 ∈ W . Mrigendra Singh Kushwaha A path approach to Kostant modules

Filtration of V ( λ ) ⊗ V ( µ ) by Kostant modules Observation 1. Let W λ and W µ be the stabilizers of dominant inte- gral weights λ and µ respectively. Then K ( λ, w 1 , µ ) = K ( λ, w 2 , µ ) if W λ w 1 W µ = W λ w 2 W µ for w 1 , w 2 ∈ W . Observation 2. Let w 1 , w 2 ∈ W , then K ( λ, w 1 , µ ) ⊆ K ( λ, w 2 , µ ) if W λ w 1 W µ ≤ W λ w 2 W µ in the Bruhat order on W λ \ W / W µ . Mrigendra Singh Kushwaha A path approach to Kostant modules

Filtration of V ( λ ) ⊗ V ( µ ) by Kostant modules Observation 1. Let W λ and W µ be the stabilizers of dominant inte- gral weights λ and µ respectively. Then K ( λ, w 1 , µ ) = K ( λ, w 2 , µ ) if W λ w 1 W µ = W λ w 2 W µ for w 1 , w 2 ∈ W . Observation 2. Let w 1 , w 2 ∈ W , then K ( λ, w 1 , µ ) ⊆ K ( λ, w 2 , µ ) if W λ w 1 W µ ≤ W λ w 2 W µ in the Bruhat order on W λ \ W / W µ . Thus we see that Kostant modules form an increasing filtration of V ( λ ) ⊗ V ( µ ) by U g -submodules, indexed by the double coset space W λ \ W / W µ thought of as a poset under the Bruhat order. Mrigendra Singh Kushwaha A path approach to Kostant modules

Some basic notions and results due to Littelmann Let B λ be the set of Lakshmibai-Seshadri(L-S) paths of shape λ . Recall that a path π ∈ B λ consists of a sequence τ 1 > τ 2 > ... > τ r of elements of W / W λ and a sequence of rational numbers 0 = a 0 < a 1 < ... < a r = 1. We call τ 1 the initial direction and τ r the final direction of π. Mrigendra Singh Kushwaha A path approach to Kostant modules

Some basic notions and results due to Littelmann Let B λ be the set of Lakshmibai-Seshadri(L-S) paths of shape λ . Recall that a path π ∈ B λ consists of a sequence τ 1 > τ 2 > ... > τ r of elements of W / W λ and a sequence of rational numbers 0 = a 0 < a 1 < ... < a r = 1. We call τ 1 the initial direction and τ r the final direction of π. Root operators For every simple root α , Littelmann associated two operators e α and f α on the set of paths. Let B λ ∗ B µ := { π ∗ π ′ | π ∈ B λ , π ′ ∈ B µ } , where ∗ denotes con- catenation, and µ ∈ Λ + . Mrigendra Singh Kushwaha A path approach to Kostant modules

Kostant Set Given a path π ∗ π ′ ∈ B λ ∗ B µ , we associate a Weyl group element m ( π ∗ π ′ ) by: m ( π ∗ π ′ ) := min W λ I ( τ − 1 ) σ W µ min W λ I ( τ − 1 ) σ W µ : unique minimal element of this set (exists). τ : lift in W of the final direction of π . σ : lift in W of the initial direction of π ′ . I ( τ − 1 ) : Bruhat interval { w ∈ W | w ≤ τ − 1 } . Mrigendra Singh Kushwaha A path approach to Kostant modules

Kostant Set Kostant Set in B λ ∗ B µ Given an element φ of the double coset space W λ \ W / W µ , define the corresponding Kostant set by: ( B λ ∗ B µ ) φ := { π ∗ π ′ ∈ B λ ∗ B µ | m ( π ∗ π ′ ) ≤ � φ } where � φ is any lift of φ (the choice of lift doesn’t matter). Mrigendra Singh Kushwaha A path approach to Kostant modules

Kostant Set Kostant Set in B λ ∗ B µ Given an element φ of the double coset space W λ \ W / W µ , define the corresponding Kostant set by: ( B λ ∗ B µ ) φ := { π ∗ π ′ ∈ B λ ∗ B µ | m ( π ∗ π ′ ) ≤ � φ } where � φ is any lift of φ (the choice of lift doesn’t matter). ( B λ ∗ B µ ) φ ⊆ ( B λ ∗ B µ ) φ ′ if φ ≤ φ ′ . Kostant sets form an increasing filtration of B λ ∗ B µ indexed by the Bruhat poset W λ \ W / W µ . Mrigendra Singh Kushwaha A path approach to Kostant modules

Stability of Kostant set under root operators Lemma [ , Raghavan,Viswanath, 2018] Let π ∗ π ′ and σ ∗ σ ′ be paths in B λ ∗ B µ such that σ ∗ σ ′ equals either e α ( π ∗ π ′ ) or f α ( π ∗ π ′ ) for some simple root α. Then m ( π ∗ π ′ ) = m ( σ ∗ σ ′ ) . Mrigendra Singh Kushwaha A path approach to Kostant modules

Stability of Kostant set under root operators Lemma [ , Raghavan,Viswanath, 2018] Let π ∗ π ′ and σ ∗ σ ′ be paths in B λ ∗ B µ such that σ ∗ σ ′ equals either e α ( π ∗ π ′ ) or f α ( π ∗ π ′ ) for some simple root α. Then m ( π ∗ π ′ ) = m ( σ ∗ σ ′ ) . Equivalence relation on B λ ∗ B µ Let π ∗ π ′ and σ ∗ σ ′ be the paths in B λ ∗ B µ , let us say π ∗ π ′ related to σ ∗ σ ′ , if π ∗ π ′ equals either e α ( σ ∗ σ ′ ) or f α ( σ ∗ σ ′ ). This relation is symmetric since π ∗ π ′ = e α ( σ ∗ σ ′ ) if and only if f α ( π ∗ π ′ ) = σ ∗ σ ′ . Denote by ∼ the reflexive and transitive closure of this relation (as α varies over all simple roots). Thus a Kostant set is a union of equivalence classes of ∼ . Mrigendra Singh Kushwaha A path approach to Kostant modules

Path model for the Kostant module Given φ ∈ W λ \ W / W µ , consider the Kostant set ( B λ ∗ B µ ) φ := { π ∗ π ′ ∈ B λ ∗ B µ | m ( π ∗ π ′ ) ≤ � φ } Mrigendra Singh Kushwaha A path approach to Kostant modules

Path model for the Kostant module Given φ ∈ W λ \ W / W µ , consider the Kostant set ( B λ ∗ B µ ) φ := { π ∗ π ′ ∈ B λ ∗ B µ | m ( π ∗ π ′ ) ≤ � φ } Theorem [ , Raghavan,Viswanath, 2018] Let g be a finite dimensional complex semisimple Lie algebra. ( B λ ∗ B µ ) φ is a path model for the Kostant module K ( λ, φ, µ ), i.e., � e π (1) char K ( λ, φ, µ ) = char ( B λ ∗ B µ ) φ := π ∈ ( B λ ∗ B µ ) φ Mrigendra Singh Kushwaha A path approach to Kostant modules

Decomposition rule for Kostant modules Theorem [ , Raghavan,Viswanath, 2018] Let g be a finite dimensional complex semisimple Lie algebra. Let λ, µ be dominant integral weights and w an element of Weyl group. The decomposition of the Kostant module K ( λ, w , µ ) as a direct sum of irreducible g -modules is given by � K ( λ, w , µ ) ∼ = V ( λ + π (1)) π ∈ B λ µ ( w ) where B λ µ := { π ∈ B µ | λ + π ( t ) dominant for all t ∈ [0 , 1] } and B λ µ ( w ) := { π ∈ B λ µ | initial direction of π is ≤ wW µ } . Mrigendra Singh Kushwaha A path approach to Kostant modules

Decomposition rule for Kostant modules Theorem [ , Raghavan,Viswanath, 2018] Let g be a finite dimensional complex semisimple Lie algebra. Let λ, µ be dominant integral weights and w an element of Weyl group. The decomposition of the Kostant module K ( λ, w , µ ) as a direct sum of irreducible g -modules is given by � K ( λ, w , µ ) ∼ = V ( λ + π (1)) π ∈ B λ µ ( w ) where B λ µ := { π ∈ B µ | λ + π ( t ) dominant for all t ∈ [0 , 1] } and B λ µ ( w ) := { π ∈ B λ µ | initial direction of π is ≤ wW µ } . Note: Proof uses Kumar’s [Inv. Math, 1988] character formula for the Kostant module and follows Littelmann’s [Inv. Math, 1994] proof of the decomposition rule for V ( λ ) ⊗ V ( µ ). Mrigendra Singh Kushwaha A path approach to Kostant modules

PRV and refinements Let ν = λ + σµ (dominant conjugate) for some σ ∈ W . Mrigendra Singh Kushwaha A path approach to Kostant modules