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 vλ be a highest weight vector in V (λ). Let vwµ be a non-zero vector in the (one-dimensional) weight space

• f weight wµ 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 vλ ⊗ vwµ: K(λ, w, µ) := Ug(vλ ⊗ vwµ) where Ug 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 vλ be a highest weight vector in V (λ). Let vwµ be a non-zero vector in the (one-dimensional) weight space

• f weight wµ 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 vλ ⊗ vwµ: K(λ, w, µ) := Ug(vλ ⊗ vwµ) where Ug denotes the universal enveloping algebra of g. Examples w = 1 : K(λ, 1, µ) ∼ = V (λ + µ). w = w0 the longest element ( if g is finite dimensional ) : K(λ, w0, 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 Wλ and Wµ be the stabilizers of dominant inte- gral weights λ and µ respectively. Then K(λ, w1, µ) = K(λ, w2, µ) if Wλ w1 Wµ = Wλ w2 Wµ for w1, w2 ∈ W .

Mrigendra Singh Kushwaha A path approach to Kostant modules

Filtration of V (λ) ⊗ V (µ) by Kostant modules

Observation 1. Let Wλ and Wµ be the stabilizers of dominant inte- gral weights λ and µ respectively. Then K(λ, w1, µ) = K(λ, w2, µ) if Wλ w1 Wµ = Wλ w2 Wµ for w1, w2 ∈ W . Observation 2. Let w1, w2 ∈ W , then K(λ, w1, µ) ⊆ K(λ, w2, µ) if Wλ w1 Wµ ≤ Wλ w2 Wµ in the Bruhat order on Wλ\W /Wµ.

Mrigendra Singh Kushwaha A path approach to Kostant modules

Filtration of V (λ) ⊗ V (µ) by Kostant modules

Observation 1. Let Wλ and Wµ be the stabilizers of dominant inte- gral weights λ and µ respectively. Then K(λ, w1, µ) = K(λ, w2, µ) if Wλ w1 Wµ = Wλ w2 Wµ for w1, w2 ∈ W . Observation 2. Let w1, w2 ∈ W , then K(λ, w1, µ) ⊆ K(λ, w2, µ) if Wλ w1 Wµ ≤ Wλ w2 Wµ in the Bruhat order on Wλ\W /Wµ. Thus we see that Kostant modules form an increasing filtration of V (λ)⊗V (µ) by Ug-submodules, indexed by the double coset space Wλ\W /Wµ 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 Bλ be the set of Lakshmibai-Seshadri(L-S) paths of shape λ. Recall that a path π ∈ Bλ consists of a sequence τ1 > τ2 > ... > τr

• f elements of W /Wλ and a sequence of rational numbers 0 = a0 <

a1 < ... < ar = 1. We call τ1 the initial direction and τr the final direction of π.

Mrigendra Singh Kushwaha A path approach to Kostant modules

Some basic notions and results due to Littelmann

Let Bλ be the set of Lakshmibai-Seshadri(L-S) paths of shape λ. Recall that a path π ∈ Bλ consists of a sequence τ1 > τ2 > ... > τr

• f elements of W /Wλ and a sequence of rational numbers 0 = a0 <

a1 < ... < ar = 1. We call τ1 the initial direction and τr the final direction of π. Root operators For every simple root α, Littelmann associated two operators eα and fα on the set of paths. Let Bλ ∗ Bµ := {π ∗ π′|π ∈ Bλ, π′ ∈ Bµ}, where ∗ denotes con- catenation, and µ ∈ Λ+.

Mrigendra Singh Kushwaha A path approach to Kostant modules

Kostant Set

Given a path π ∗ π′ ∈ Bλ ∗ Bµ, we associate a Weyl group element m(π ∗ π′) by: m(π ∗ π′) := min Wλ I(τ −1) σ Wµ min Wλ I(τ −1) σ Wµ : 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 Bλ ∗ Bµ Given an element φ of the double coset space Wλ\W /Wµ, define the corresponding Kostant set by: (Bλ ∗ Bµ)φ := { π ∗ π′ ∈ Bλ ∗ Bµ | 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 Bλ ∗ Bµ Given an element φ of the double coset space Wλ\W /Wµ, define the corresponding Kostant set by: (Bλ ∗ Bµ)φ := { π ∗ π′ ∈ Bλ ∗ Bµ | m(π ∗ π′) ≤ φ } where φ is any lift of φ (the choice of lift doesn’t matter). (Bλ ∗ Bµ)φ ⊆ (Bλ ∗ Bµ)φ′ if φ ≤ φ′. Kostant sets form an increasing filtration of Bλ ∗ Bµ indexed by the Bruhat poset Wλ\W /Wµ.

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 Bλ∗Bµ such that σ∗σ′ equals either eα(π ∗ π′) or fα(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 Bλ∗Bµ such that σ∗σ′ equals either eα(π ∗ π′) or fα(π ∗ π′) for some simple root α. Then m(π ∗ π′) = m(σ ∗ σ′). Equivalence relation on Bλ ∗ Bµ Let π ∗ π′ and σ ∗ σ′ be the paths in Bλ ∗ Bµ, let us say π ∗ π′ related to σ ∗ σ′, if π ∗ π′ equals either eα(σ ∗ σ′) or fα(σ ∗ σ′). This relation is symmetric since π ∗ π′ = eα(σ ∗ σ′) if and only if fα(π ∗ π′) = σ ∗ σ′. Denote by ∼ the reflexive and transitive closure

• f 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 ∈ Wλ\W /Wµ, consider the Kostant set (Bλ ∗ Bµ)φ := {π ∗ π′ ∈ Bλ ∗ Bµ| m(π ∗ π′) ≤ φ}

Mrigendra Singh Kushwaha A path approach to Kostant modules

Path model for the Kostant module

Given φ ∈ Wλ\W /Wµ, consider the Kostant set (Bλ ∗ Bµ)φ := {π ∗ π′ ∈ Bλ ∗ Bµ| m(π ∗ π′) ≤ φ} Theorem [ , Raghavan,Viswanath, 2018] Let g be a finite dimensional complex semisimple Lie algebra. (Bλ ∗Bµ)φ is a path model for the Kostant module K(λ, φ, µ), i.e., char K(λ, φ, µ) = char (Bλ ∗ Bµ)φ :=

• π∈(Bλ∗Bµ)φ

eπ(1)

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) ∼ =

• π∈Bλ

µ(w)

V (λ + π(1)) where Bλ

µ := {π ∈ Bµ| λ + π(t) dominant for all t ∈ [0, 1]} and

Bλ

µ(w) := {π ∈ Bλ µ| initial direction of π is ≤ wWµ}.

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) ∼ =

• π∈Bλ

µ(w)

V (λ + π(1)) where Bλ

µ := {π ∈ Bµ| λ + π(t) dominant for all t ∈ [0, 1]} and

Bλ

µ(w) := {π ∈ Bλ µ| initial direction of π is ≤ wWµ}.

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

PRV and refinements

Let ν = λ + σµ (dominant conjugate) for some σ ∈ W . PRV conjecture V (ν) occurs with multiplicity ≥ 1 in 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 . PRV conjecture V (ν) occurs with multiplicity ≥ 1 in V (λ) ⊗ V (µ). Kostant’s refinement [Kumar 1988] V (ν) occurs with multiplicity ≥ 1 in K(λ, w, µ) ⊆ V (λ) ⊗ V (µ) for all Wλ w Wµ ≥ Wλ σ Wµ (exactly 1 for σ = w).

Mrigendra Singh Kushwaha A path approach to Kostant modules

PRV and refinements

Let ν = λ + σµ (dominant conjugate) for some σ ∈ W . Verma’s refinement [Kumar 1989] In V (λ) ⊗ V (µ), V (ν) occurs with multiplicity ≥ # double cosets Wλ τ Wµ such that ν = λ + τµ.

Mrigendra Singh Kushwaha A path approach to Kostant modules

PRV and refinements

Let ν = λ + σµ (dominant conjugate) for some σ ∈ W . Verma’s refinement [Kumar 1989] In V (λ) ⊗ V (µ), V (ν) occurs with multiplicity ≥ # double cosets Wλ τ Wµ such that ν = λ + τµ. Kostant-Verma simultaneous refinement [ , Raghavan,Viswanath, 2018] In K(λ, w, µ), V (ν) occurs with multiplicity ≥ # double cosets Wλ τ Wµ such that ν = λ + τµ and Wλ τ Wµ ≤ Wλ w Wµ (w = w0 gives Verma’s refinement).

Mrigendra Singh Kushwaha A path approach to Kostant modules

Another consequence

Proposition [Kumar 1988] Suppose λ, µ ∈ Λ+ are both regular. Fix w ∈ W . Then the g-module V (λ + wµ) does not occur in K(λ, v, µ), for any v < w. Proposition [ , Raghavan,Viswanath, 2018] Let λ, µ ∈ Λ+ and fix Wλ w Wµ ∈ Wλ\W /Wµ. Then the g-module V (λ + wµ) does not occur in K(λ, v, µ), for any Wλ v Wµ < Wλ w Wµ.

Mrigendra Singh Kushwaha A path approach to Kostant modules

Another consequence

Proposition [Kumar 1988] Suppose λ, µ ∈ Λ+ are both regular. Fix w ∈ W . Then the g-module V (λ + wµ) does not occur in K(λ, v, µ), for any v < w. Proposition [ , Raghavan,Viswanath, 2018] Let λ, µ ∈ Λ+ and fix Wλ w Wµ ∈ Wλ\W /Wµ. Then the g-module V (λ + wµ) does not occur in K(λ, v, µ), for any Wλ v Wµ < Wλ w Wµ.

Thank You

Mrigendra Singh Kushwaha A path approach to Kostant modules