Yang-Baxter basis of Hecke algebra and Casselman's problem (extended abstract)

We generalize the definition of Yang-Baxter basis of type $A$ Hecke algebra introduced by A.Lascoux, B.Leclerc and J.Y.Thibon (Letters in Math. Phys., 40 (1997), 75--90) to all the Lie types and prove their duality. As an application we give a solution to Casselman's problem on Iwahori fixed vectors of principal series representation of $p$-adic groups.


Introduction
Yang-Baxter basis of Hecke algebra of type A was defined in the paper of Lascoux-Leclerc-Thibon [LLT]. There is also a modified version in [Las]. First we generalize the latter version to all the Lie types. Then we will solve the Casselman's problem on the basis of Iwahori fixed vectors using Yang-Baxter basis and Demazure-Lusztig type operator. This paper is an extended abstract and the detailed proofs will appear in [NN].
2 Generic Hecke algebra 2.1 Root system, Weyl group and generic Hecke algebra Let R = (Λ, Λ * , R, R * ) be a (reduced) semisimple root data cf. [Dem]. More precisely Λ ≃ Z r is a weight lattice with rank Λ = r. There is a pairing < , >: Λ * × Λ → Z. R ⊂ Λ is a root system with simple roots {α i } 1≤i≤r and positive roots R + . R * ⊂ Λ * is the set of coroots, and there is a bijection R → R * , α → α * . We also denote the coroot α * = h α . The Weyl group W of R is generated by simple reflections S = {s i } 1≤i≤r . The action of W on Λ is given by s i (λ) = λ− < α * i , λ > α i for λ ∈ Λ. We define generic Hecke algebra H t1,t2 (W ) over Z[t 1 , t 2 ] with two parameters t 1 , t 2 as follows. Generators are h i = h si , with relations (h i − t 1 )(h i − t 2 ) = 0 for 1 ≤ i ≤ r and the braid relations h i h j · · · mi,j = h j h i · · · mi,j , where m i,j is the order of s i s j for 1 ≤ i < j ≤ r.
We need to extend the coefficients to the quotient field of the group algebra Z [Λ]. An element of Z[Λ] is denoted as λ∈Λ c λ e λ . The Weyl group acts on Z [Λ] by w(e λ ) = e wλ . We extend the coefficient ring Z[t 1 , t 2 ] of H t1,t2 (W ) to For w ∈ W , an expression of w = s i1 s i2 · · · s i ℓ with minimal number of generators s i k ∈ S is called a reduced expression in which case we write ℓ(w) = ℓ and call it the length of w.

Yang-Baxter basis and its properties
Yang-Baxter basis was introduced in the paper [LLT] to investigate the relation with Schubert calculus. There is also a variant in [Las] for type A case. We generalize that results to all Lie types.
Proof. We can prove these equations by direct calculations.
Remark 1. In [Che] I. Cherednik treated Yang-Baxter relation in more general setting. There is also a related work [Kat] by S. Kato and the proof of Theorem 2.4 in [Kat] suggests a uniform way to prove Yang-Baxter relations without direct calculations.
We use the Bruhat order x ≤ y on elements x, y ∈ W (cf. [Hum]). Following [Las] we define the Yang-Baxter basis Y w for w ∈ W recursively as follows.
Using the Yang-Baxter relation above it is easy to see that Y w does not depend on a reduced expression of w. As the leading term of Y w with respect to the Bruhat order is h w , they also form a Q t1,t2 (Λ)-basis {Y w } w∈W of H Q(Λ) t1,t2 (W ).
We are interested in the transition coefficients p(w, v) andp(w, v) ∈ Q t1,t2 (Λ) between the two basis {Y w } w∈W and {h w } w∈W , i.e.
Take a reduced expression of v e.g. v = s i1 · · · s i ℓ where ℓ = ℓ(v) is the length of v (cf. [Hum]). Then Y v is expressed as follows.
Remark 2. The relation to K-theory Schubert calculus is as follows. If we set t 1 = 0, t 2 = −1 and replacing at v of the equivariant K-theory Schubert class ψ w (cf. [LSS]).
Proposition 2. (Lascoux [Las] Lemma 1.8.1 for type A case) For v ∈ W , where W acts only on the coefficients.
Proof. When ℓ(v) > 0 there exists s ∈ S such that v = v ′ s > v ′ . Using the induction assumption on v ′ , we get the formula for v.
Taking the coefficient of h w in the above equation, we get

Inner product and orthogonality
t1,t2 (W ) and s ∈ S. There is an involutionˆ: H The following proposition is due to A.Lascoux for the type A case [Las] P.33.
Proof. We can use induction on the length ℓ(v) of v to prove the equation.
We have another orthogonality between Y v and w 0 (Y w0w ).
Proposition 4. (Type A case was due to [LLT] Theorem 5.1 , [Las] Theorem Proof. We use induction on ℓ(v) and use the fact that if s ∈ S and u ∈ W , then Y u h s = aY us + bY s for some a, b ∈ Q t1,t2 (Λ).

Duality between the transition coefficients
Recall that we have two transition coefficients p(w, v),p(w, v) ∈ Q t1,t2 (Λ) defined by the following expansions.
Below gives a relation between them.
Theorem 1. (Lascoux [Las] Corollary 1.8.5 for type A case) For w, v ∈ W , by the orthogonality on Y v (Proposition 4). On the other hand, as Then using the orthogonality on h v (Proposition 3) and Corollary 1, The theorem is proved.

Recurrence relations
Here we give some recurrence relations on p(w, v) andp(w, v).
We note that by this recurrence we can identify p(w, v) as a coefficient of transition between two bases of the space of Iwahori fixed vectors cf. Theorem 3 below.
and taking the coefficient of h w , we get the formula.
Proof. We can prove the recurrence relation using Corollary 2 below.
Proof. We can prove the recurrence relation using Corollary 2 below.

Kostant-Kumar's twisted group algebra
be the (generic) twisted group algebra of Kostant-Kumar. Its element is of the form w∈W f w δ w for f w ∈ Q t1,t2 (Λ) and the product is defined by Define y i ∈ Q KK t1,t2 (W ) (i = 1, . . . , r) by Proposition 9. We have the following equations.
(2) y i y j · · · mi,j = y j y i · · · mi,j , where m i,j is the order of s i s j .
Proof. These equations can be shown by direct calculations.
By this proposition we can define y w := y i1 · · · y i ℓ for a reduced expression w = s i1 · · · s i ℓ . These {y w } w∈W become a Q t1,t2 (Λ)-basis of Q KK t1,t2 (W ).
Remark 3. This operator y i can be seen as a generic Demazure-Lusztig operator. When t 1 = −1, t 2 = q, it becomes y q si in Kumar's book [Kum](12.2.E(9)). We can also set A i which satisfies For example, if we set A i = t1+t2e α i 1−e α i and t 1 = q, t 2 = −1 and replace α i by −α i , it becomes Lusztig's T si [Lu1]. If we set A i = − t1+t2e α i 1−e −α i and t 1 = −1, t 2 = v and replace α i by −α i , it becomes T i in [BBL].
Theorem 2. For w ∈ W , we have We used the recurrence relation (Proposition 5) for the last equality. Therefore Φ(∆ siw ) = u≤siw p(u, s i w)h u = Y siw . The theorem is proved.

Proof. Taking the inverse image of the map Φ, the equality h
As v = s i1 · · · s i ℓ is a reduced expression, y v = y si 1 · · · y si ℓ = (A ii δ i1 + B i1 δ e ) · · · (A i ℓ δ i ℓ + B i ℓ δ e ). By expanding this we get the formula.
Remark 4. Using Theorem 1, we also have a closed form for p(w, v). We have another conjectural formula for p(w, v) using λ-chain cf. [Nar].

Casselman's problem
In his paper [Cas] B. Casselman gave a problem concerning transition coefficients between two bases in the space of Iwahori fixed vectors of a principal series representation of a p-adic group. We relate the problem with the Yang-Baxter basis and give an answer to the problem.

Principal series representations of p-adic group and Iwahori fixed vector
We follow the notations of M.Reeder [Re1,Re2]. Let G be a connected reductive p-adic group over a non-archimedian local field F . For simplicity we restrict to the case of split semisimple G. Associated to F , there is the ring of integer O, the prime ideal p with a generator ̟, and the residue field with q = |O/p| elements. Let P be a minimal parabolic subgroup (Borel) of G, and A be the maximal split torus of P so that A ≃ (F * ) r where r is the rank of G. For an unramified quasi-character τ of A, i.e. a group homomorphism τ : We have a pairing <, >: A/A 0 × T → C * given by < a, z ⊗ λ >= z val(λ(a)) . This gives an identification T ≃ X nr (A) of T with the set of unramified quasicharacters on A (cf. [Bum] Exercise 18,19). Let ∆ ⊂ X * (A) be the set of roots of A in G, ∆ + be the set of positive roots corresponding to P and Σ ⊂ ∆ + be the set of simple roots . For a root α ∈ ∆, we define e α ∈ X * (T ) by e α (τ ) =< h α (̟), τ > for τ ∈ T where h α : F * → A is the one parameter subgroup (coroot) corresponding to α.
Remark 5. As the definition shows, e α is defined using the coroot α * = h α . So it should be parametrized by α * , but for convenience we follow the notation of [Re1]. Later we will identify e α (α ∈ ∆ = R * ) with e α (α ∈ R = ∆ * ) by the map * : ∆ → R of root data.
The principal series representation I(τ ) of G associated to a unramified quasicharacter τ of A is defined as follows. As a vector space over C it consists of locally constant functions on G with values in C which satisfy the left relative invariance properties with respect to P where τ is extended to P with trivial value on the unipotent radical N of P = AN . I(τ ) := Ind G P (τ ) = {f : G → C loc. const. function |f (pg) = τ δ 1/2 (p)f (g) for ∀p ∈ P, ∀g ∈ G}.
Here δ is the modulus of P . The action of G on I(τ ) is defined by right translation, i.e. for g ∈ G and f ∈ I(τ ), (π(g)f )(x) = f (xg). Let B be the Iwahori subgroup which is the inverse image π −1 (P (F q )) of the Borel subgroup P (F q ) of G(F q ) by the projection π : G(O) → G(F q ). Then we define I(τ ) B to be the space of Iwahori fixed vectors in I(τ ), i.e.

Intertwiner and Casselman's basis
From now on we always assume that τ is regular i.e. the stabilizer with N − being the unipotent radical of opposite parabolic P − which corresponds to the negative roots ∆ − . The integral is convergent when |e α (τ )| < 1 for all α ∈ ∆ + such that wα ∈ ∆ − (cf. [Bum] Proposition 63), and may be meromorphically continued to all τ . It has the property that for x, y ∈ W with ℓ(xy) = ℓ(x) + ℓ(y), then The Casselman's basis {f τ w } w∈W of I(τ ) B is defined as follows. f τ w ∈ I(τ ) B and M.Reeder characterizes this using the action of affine Hecke algebra (cf. [Re2] Section 2). The affine Hecke algebra H = H(G, B) is the convolution algebra of B bi-invariant locally constant functions on G with values in C. By the theorem of Iwahori-Matsumoto it can be described by generators and relations. The basis {T w } w∈ W af f consists of characteristic functions T w := ch BwB of double coset BwB. Let H W be the Hecke algebra of the finite Weyl group W generated by the simple reflections s α for simple roots α ∈ Σ. As a vector space H is the tensor product of two subalgebras H = Θ ⊗ H W . The subalgebra Θ is commutative and isomorphic to the coordinate ring of the complex torus T with a basis {θ a | a ∈ A/A 0 }, where θ a is defined as follows (cf. [Lu2]). Define A − := {a ∈ A | |α(a)| F ≤ 1 ∀α ∈ Σ}. For a ∈ A, choose a 1 , a 2 ∈ A − such that a = a 1 a −1 2 . Then θ a = q (ℓ(a1)−ℓ(a2))/2 T a1 T −1 a2 where for x ∈ G, ℓ(x) is the length function defined by q ℓ(x) = [BxB : B] and T x ∈ H is the characteristic function of BxB.

Transition coefficients
The Casselman's problem is to find an explicit formula for a w,v (τ ) and b w,v (τ ).
To relate the results in Sections 2 and 3 with the Casselman's problem, in this subsection we specialize the parameters t 1 = −q −1 , t 2 = 1 and take tensor product with the complex field C. For example, the Yang-Baxter basis Y w will become a Q t1,t2 (Λ) ⊗ C basis in H The generic Demazure-Lusztig operator defined in Section 3 will become Then (y i + q −1 )(y i − 1) = 0.
Theorem 3. We identify e α with e α (cf. Remark 4). Then, Proof. b w,v 's satisfy the same recurrence relation (Proposition 5 with t 1 = −q −1 , t 2 = 1) as p(w, v)'s (cf. [Re2] Proposition (2.2)). The initial condition b w,w = p(w, w) = 1 leads to the second equation. The first equation then also holds. Note that the b y,w in [Re2] is our b w,y .
Remark 6. There is also a direct proof that does not use recurrence relation cf. [NN].
Corollary 3. We have a closed formula for a w,v (τ ) and b w,v (τ ) by Corollary 2 and Theorem1.
Remark 7. The left hand side of the first equation in Corollary 4 is m(e, v −1 ) in [BN]. So this gives another proof of Theorem 1.4 in [BN].
Formally the result of M.Reeder [Re2] Corollary (3.2) is written as follows. For w ∈ W and a ∈ A − , W(ϕ w )(a) = δ 1/2 (a) w≤y b w,y y   λ a β∈R + −R(y) Then using Corollary 3, we have an explicit formula of W(ϕ w )(a).

Relation with Bump-Nakasuji's work
Now we explain the relation between this paper and Bump-Nakasuji [BN]. First of all, the notational conventions are slightly different. Especially in the published [BN] the natural base and intertwiner are differently parametrized. The natural basis φ w in [BN] is our ϕ w −1 . The intertwiner M w in [BN] is our A w −1 so that if ℓ(w 1 w 2 ) = ℓ(w 1 ) + ℓ(w 2 ), M w1w2 = M w1 • M w2 while A w1w2 = A w2 A w1 .