## Orthogonality and formulae for Kostka–Foulkes polynomials

Last update: 27 September 2012

## Orthogonality and formulae for Kostka–Foulkes polynomials

Let $𝕂=ℤ\left[t\right]\text{.}$ If $f=\sum _{\mu \in P}{f}_{\mu }{x}^{\mu }\in 𝕂\left[P\right]$ let

$f‾=∑μ∈P fμx-μ,and [f]1=f0= (coefficient of 1 inf). (3.1)$

Define a symmetric bilinear form

$⟨,⟩t: 𝕂[P]×𝕂[P] ⟶𝕂by ⟨f,g⟩t= 1∣W∣ [ fg‾ ∏α∈R 1-xα 1-txα ] 1 . (3.2)$

"Specializing" $t$ at the values 0 and 1 gives inner products

$⟨,⟩0: 𝕂[P]×𝕂[P] ⟶𝕂and ⟨,⟩1: 𝕂[P]×𝕂[P] ⟶𝕂$

with

$⟨f,g⟩0= 1∣W∣ [ fg‾ ∏α∈R (1-xα) ] 1 and ⟨f,g⟩1= 1∣W∣ [fg‾]1. (3.3)$

Let $\lambda$ and $\mu \in {P}^{+}\text{.}$ Then

$⟨mλ,mμ⟩1 =1∣Wλ∣ δλμ, ⟨sλ,sμ⟩0 =δλμ,and ⟨Pλ,Pμ⟩t =1Wλ(t) δλμ.$

 Proof. Letting $W\lambda$ denote the $W$-orbit of $\lambda ,$ the first equality follows from $∣Wλ∣ ⟨mλ,mμ⟩1 =∣Wλ∣∣W∣ ∑γ∈Wλ,ν∈Wμ [xγx-ν]1= δλμ ∣Wλ∣∣W∣ ∑γ∈Wλ1= δλμ.$ If $\lambda ,\mu \in {P}^{+},$ $⟨sλ,sμ⟩0 = 1∣W∣ [aρsλ‾aρsμ]1 =1∣W∣ [aλ+ρ‾aμ+ρ]1 = 1∣W∣ ∑v,w∈W (-1)ℓ(v) (-1)ℓ(w) [ x-v(λ+ρ) xw(μ+ρ) ] 1 = δλμ 1∣W∣ ∑v∈W (-1)ℓ(v) (-1)ℓ(v) =δλμ,$ giving the second statement. By Lemma 2.5(b) the matrix ${K}^{-1}$ given by the values ${\left({K}^{-1}\right)}_{\lambda \mu }$ in the equation $Pλ(x;t)=∑μ (K-1)λμ sμ,$ has entries in $ℤ\left[t\right]$ and is upper triangular with 1's on the diagonal, that is, ${\left({K}^{-1}\right)}_{\lambda \lambda }=1$ and ${\left({K}^{-1}\right)}_{\lambda \mu }=0$ unless $\mu \le \lambda \text{.}$ Since ${P}_{\lambda }\left(x;1\right)={m}_{\lambda }$ the matrix ${k}^{-1}$ describing the change of basis $mλ=∑μ (k-1)λμ sμ,$ is the specialization of ${K}^{-1}$ at $t=1$ and so ${k}^{-1}$ has entries in $ℤ$ and is upper triangular with 1's on the diagonal. Then the matrix $A={K}^{-1}{k}^{-1}$ giving the change of basis $Pλ(x;t)= ∑ν≤λ Aλνmμ, (3.4)$ has ${A}_{\lambda \mu }\in ℤ\left[t\right],$ ${A}_{\lambda \lambda }=1,$ and ${A}_{\lambda \mu }=0$ unless $\mu \le \lambda \text{.}$ Let ${Q}^{+}$ be the set of nonnegative integral linear combinations of positive roots. Then $Pμ(x;t) Wμ(t) ( ∏α∈R 1-xα 1-txα ) = ∑w∈Ww ( xμ ∏α∈R+ 1-xα 1-txα ) = ∑w∈Ww ( xμ ∏α∈R+ ( 1+∑r>0 tr-1 (t-1) xrα ) ) = ∑w∈Ww ( ∑ν∈Q+ cν xμ+ν ) = ∑ν∈Q+ cν ( ∑w∈W wxμ+ν ) ,$ where ${c}_{\nu }\in ℤ\left[t\right]$ and ${c}_{0}=1\text{.}$ Hence $Pμ(x;t) Wμ(t) ∏α∈R 1-xα 1-txα =∣Wμ∣mμ+ ∑γ>μ Bμγmγ= ∑γ≥μ Bμγmγ, (3.5)$ with ${B}_{\mu \gamma }\in ℤ\left[t\right]$ and ${B}_{\mu \mu }=\mid {W}_{\mu }\mid \text{.}$ Assume that $\lambda \le \mu$ if $\lambda$ and $\mu$ are comparable. Then, by using (3.4) and (3.5), $⟨Pλ,Pμ⟩ t = 1Wμ(t) ⟨ Pλ,PμWμ (t)∏α∈R 1-xα 1-txα ⟩ 1 = 1Wμ(t) ⟨ ∑ν≤λ Aλνmν, ∑γ≥μ Bμγmγ ⟩ 1 .$ Since ${A}_{\lambda \lambda }=1$ and ${B}_{\mu \mu }=\mid {W}_{\mu }\mid$ the result follows from ${⟨{m}_{\lambda },{m}_{\mu }⟩}_{1}={\mid {W}_{\lambda }\mid }^{-1}{\delta }_{\lambda \mu }\text{.}$ $\square$

The following theorem shows that the spherical functions ${P}_{\lambda }\left(x;t\right)$ are uniquely determined by the triangularity in (3.4) and the orthogonality in the third equality of Proposition 3.1.

Let $𝕂=ℤ\left[t\right]\text{.}$ The spherical functions ${P}_{\lambda }\left(x;t\right)$ are the unique elements of $𝕂{\left[P\right]}^{W}$ such that

1. ${P}_{\lambda }={m}_{\lambda }+\sum _{\mu <\lambda }{A}_{\lambda \mu }{m}_{\mu },$
2. ${⟨{P}_{\lambda },{P}_{\mu }⟩}_{t}=0\phantom{\rule{0.2em}{0ex}}\text{if}\phantom{\rule{0.2em}{0ex}}\lambda \ne \mu \text{.}$

 Proof. Assume that the ${P}_{\mu }$ are determined for $\mu <\lambda \text{.}$ Then the condition in (a) can be rewritten as $Pλ=mλ+ ∑μ<λ CλμPμ,$ for some constants ${C}_{\lambda \mu }\text{.}$ Take the inner product on each side with ${P}_{\nu },$ $\nu <\lambda ,$ and use property (b) to get the system of equations $0=⟨mλ,Pν⟩t +∑μ<λCλμ ⟨Pμ,Pν⟩t =⟨mλ,Pν⟩t +Cλν ⟨Pν,Pν⟩t.$ Hence $Cλν= -⟨mλ,Pν⟩t ⟨Pν,Pν⟩t ,for eachν<λ,$ and this determines ${P}_{\lambda }\text{.}$ $\square$

(a) The inner product ${⟨,⟩}_{t}$ arises naturally in the context of $p$-adic groups. Let ${S}^{1}=\left\{z\in ℂ\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}\mid z\mid =1\right\}$ and view the ${x}^{\lambda },$ $\lambda \in P,$ as characters of

$T=Hom(P,S1) via xλ: T⟶C* s⟼ s(λ). (3.6)$

Let $ds$ be the Haar measure on $T$ normalized so that

$⟨xλ,xμ⟩ =∫Txλ(s) xμ(s)‾ds= δλμ. (3.7)$

Letting ${ℚ}_{p}$ be the field of $p$-adic number, Macdonald [Mac1971, (5.1.2)] showed that the Plancherel measure for the $p$-adic Chevalley group $G\left({ℚ}_{p}\right)$ corresponding to the root system $R$ is given by

$dμ(s)= W0(p-1) ∣W∣ ∏α∈R 1-xα(s) 1-p-1xα(s) . (3.8)$

The corresponding inner product is

$W0(p-1) ⟨f,g⟩ p-1 =∫Tf(s) g(s)‾dμ(s) ,forf,g∈ C(T),$

where $C\left(T\right)$ is the vector space of continuous functions on $T\text{.}$

(b) The inner product ${⟨,⟩}_{t}$ arises naturally in another representation theoretic context. The complex semisimple Lie algebra $𝔤$ corresponding to the root system $R$ acts on $S\left({𝔤}^{*}\right),$ the ring of polynomials on $𝔤,$ by the (co-)adjoint action. As graded $𝔤$-modules the characters of $S\left({𝔤}^{*}\right)$ and the subring of invariants $S{\left({𝔤}^{*}\right)}^{𝔤}$ are

$grch (S(𝔤*)) = ( ∏i=1r 11-t ) ( ∏α∈R 11-txα ) and grch (S(𝔤*)𝔤) = ∏i=1r 11-tdi= 1W0(t) ∏i=1r 11-t, (3.9)$

where $r$ is the rank of $𝔤$ and ${d}_{1},\dots ,{d}_{r}$ are the degrees of the Weyl group $W\text{.}$ Let $ℋ$ denote the vector space of harmonic polynomials. An important theorem of Kostant [Kos1963, Theorem 0.2] gives

$S(𝔤*)≅S (𝔤*)𝔤⊗ ℋ,and thus, grch(ℋ)=W0 (t)∏α∈R 11-txα. (3.10)$

If $L\left(\lambda \right)$ denotes the finite dimensional irreducible $𝔤$-module of highest weight $\lambda \in {P}^{+}$ then $L\left(\lambda \right)$ has character ${s}_{\lambda }$ and using the notation of (3.2),

$∑k≥0 dim ( Hom𝔤 ( L(λ), L(μ)⊗ ℋk ) tk ) (3.11) = ⟨ sλ,sμW0 (t)∏α∈R 11-txα ⟩ 0 = W0(t) [ sλsμ‾ ∏α∈R 1-xα 1-txα ] 1 =W0(t) ⟨sλ,sμ⟩t ,$

where ${ℋ}^{k}$ is the vector space of degree $k$ harmonic polynomials.

## Acknowledgements

The research of A. Ram was partially supported by the National Science Foundation (DMS-0097977), the National Security Agency (MDA904-01-1-0032) and by EPSRC Grant GR K99015 at the Newton Institute for Mathematical Sciences. The research of K. Nelsen was partially supported by the National Science Foundation (DMS-0097977 and a VIGRE grant) and the National Security Agency (MDA904-01-1-0032).