Kostant polynomial

In

Let Φ be a root system in a finite-dimensional real inner product space V with Weyl group W. Let Φ⁺ be a set of positive roots and Δ the corresponding set of simple roots. If α is a root, then s_α denotes the corresponding reflection operator. Roots are regarded as linear polynomials on V using the inner product α(v) = (α,v). The choice of Δ gives rise to a Bruhat order on the Weyl group determined by the ways of writing elements minimally as products of simple root reflection. The minimal length for an element s is denoted ${\displaystyle \ell (s)}$. Pick an element v in V such that α(v) > 0 for every positive root.

If α_i is a simple root with reflection operator s_i

${\displaystyle s_{i}x=x-2{(x,\alpha _{i}) \over (\alpha _{i},\alpha _{i})}\alpha _{i},}$

then the corresponding divided difference operator is defined by

${\displaystyle \delta _{i}f={f-f\circ s_{i} \over \alpha _{i}}.}$

If ${\displaystyle \ell (s)=m}$ and s has reduced expression

${\displaystyle s=s_{i_{1}}\cdots s_{i_{m}},}$

then

${\displaystyle \delta _{s}=\delta _{i_{1}}\cdots \delta _{i_{m}}}$

is independent of the reduced expression. Moreover

${\displaystyle \delta _{s}\delta _{t}=\delta _{st}}$

if ${\displaystyle \ell (st)=\ell (s)+\ell (t)}$ and 0 otherwise.

If w₀ is the longest element of W, the element of greatest length or equivalently the element sending Φ⁺ to −Φ⁺, then

${\displaystyle \delta _{w_{0}}f={\sum _{s\in W}\det s\,f\circ s \over \prod _{\alpha >0}\alpha }.}$

More generally

${\displaystyle \delta _{s}f={\det s\,f\circ s+\sum _{t<s}a_{s,t}\,f\circ t \over \prod _{\alpha >0,\,s^{-1}\alpha <0}\alpha }}$

for some constants a_s,t.

Set

${\displaystyle d=|W|^{-1}\prod _{\alpha >0}\alpha .}$

and

${\displaystyle P_{s}=\delta _{s^{-1}w_{0}}d.}$

Then P_s is a homogeneous polynomial of degree ${\displaystyle \ell (s)}$.

These polynomials are the Kostant polynomials.

Properties

Theorem. The Kostant polynomials form a free basis of the ring of polynomials over the W-invariant polynomials.

In fact the matrix

${\displaystyle N_{st}=\delta _{s}(P_{t})}$

is unitriangular for any total order such that s ≥ t implies ${\displaystyle \ell (s)\geq \ell (t)}$.

Hence

${\displaystyle \det N=1.}$

Thus if

${\displaystyle f=\sum _{s}a_{s}P_{s}}$

with a_s invariant under W, then

${\displaystyle \delta _{t}(f)=\sum _{s}\delta _{t}(P_{s})a_{s}.}$

Thus

${\displaystyle a_{s}=\sum _{t}M_{s,t}\delta _{t}(f),}$

where

${\displaystyle M=N^{-1}}$

another unitriangular matrix with polynomial entries. It can be checked directly that a_s is invariant under W.

In fact δ_i satisfies the

${\displaystyle \delta _{i}(fg)=\delta _{i}(f)g+(f\circ s_{i})\delta _{i}(g).}$

Hence

${\displaystyle \delta _{i}\delta _{s}(f)=\sum _{t}\delta _{i}(\delta _{s}(P_{t}))a_{t})=\sum _{t}(\delta _{s}(P_{t})\circ s_{i})\delta _{i}(a_{t})+\sum _{t}\delta _{i}\delta _{s}(P_{t})a_{t}.}$

Since

${\displaystyle \delta _{i}\delta _{s}=\delta _{s_{i}s}}$

or 0, it follows that

${\displaystyle \sum _{t}\delta _{s}(P_{t})\,\delta _{i}(a_{t})\circ s_{i}=0}$

so that by the invertibility of N

${\displaystyle \delta _{i}(a_{t})=0}$

for all i, i.e. a_t is invariant under W.

Steinberg basis

As above let Φ be a root system in a real inner product space V, and Φ⁺ a subset of positive roots. From these data we obtain the subset Δ = {α₁, α₂, …, α_n} of the simple roots, the coroots

${\displaystyle \alpha _{i}^{\vee }=2(\alpha _{i},\alpha _{i})^{-1}\alpha _{i},}$

and the fundamental weights λ₁, λ₂, ..., λ_n as the dual basis of the coroots.

For each element s in W, let Δ_s be the subset of Δ consisting of the simple roots satisfying s⁻¹α < 0, and put

${\displaystyle \lambda _{s}=s^{-1}\sum _{\alpha _{i}\in \Delta _{s}}\lambda _{i},}$

where the sum is calculated in the weight lattice P.

The set of linear combinations of the exponentials e^μ with integer coefficients for μ in P becomes a ring over Z isomorphic to the group algebra of P, or equivalently to the representation ring R(T) of T, where T is a maximal torus in K, the simply connected, connected compact semisimple Lie group with root system Φ. If W is the Weyl group of Φ, then the representation ring R(K) of K can be identified with R(T)^W.

Steinberg's theorem. The exponentials λ_s (s in W) form a free basis for the ring of exponentials over the subring of W-invariant exponentials.

Let ρ denote the half sum of the positive roots, and A denote the antisymmetrisation operator

${\displaystyle A(\psi )=\sum _{s\in W}(-1)^{\ell (s)}s\cdot \psi .}$

The positive roots β with sβ positive can be seen as a set of positive roots for a root system on a subspace of V; the roots are the ones orthogonal to s.λ_s. The corresponding Weyl group equals the stabilizer of λ_s in W. It is generated by the simple reflections s_j for which sα_j is a positive root.

Let M and N be the matrices

${\displaystyle M_{ts}=t(\lambda _{s}),\,\,N_{st}=(-1)^{\ell (t)}\cdot t(\psi _{s}),}$

where ψ_s is given by the weight s⁻¹ρ - λ_s. Then the matrix

${\displaystyle B_{s,s'}=\Omega ^{-1}(NM)_{s,s'}={A(\psi _{s}\lambda _{s'}) \over \Omega }}$

is triangular with respect to any total order on W such that s ≥ t implies ${\displaystyle \ell (s)\geq \ell (t)}$. Steinberg proved that the entries of B are W-invariant exponential sums. Moreover its diagonal entries all equal 1, so it has determinant 1. Hence its inverse C has the same form. Define

${\displaystyle \varphi _{s}=\sum C_{s,t}\psi _{t}.}$

If χ is an arbitrary exponential sum, then it follows that

${\displaystyle \chi =\sum _{s\in W}a_{s}\lambda _{s}}$

with a_s the W-invariant exponential sum

${\displaystyle a_{s}={A(\varphi _{s}\chi ) \over \Omega }.}$

Indeed this is the unique solution of the system of equations

${\displaystyle t\chi =\sum _{s\in W}t(\lambda _{s})\,\,a_{s}=\sum _{s}M_{t,s}a_{s}.}$

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• doi:10.1016/0040-9383(75)90025-7