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Demazure–Lusztig operators satisfy the affine type-A relations

Statement

Let $n \geq 2$ , $R_{0} = Z [Q, Q^{- 1}]$ , $A = R_{0} [X_{1}^{\pm 1}, \dots, X_{n}^{\pm 1}]$ , and $K = Frac (A)$ , with $D_{i}$ and $K ⋊ S_{n}$ as in The type-A weight lattice, Laurent ring, and divided differences and The rational skew group algebra for type A. For $r_{i} = X_{i} / X_{i + 1}$ define

T_{i} : = Q [s_{i}] + \frac{Q - 1}{1 - r_{i}} ([1] - [s_{i}]) \in K ⋊ S_{n} .

Then $T_{i}$ preserves $A$ , satisfies the quadratic and braid relations of The multiplicative Bernstein presentation of the affine Hecke algebra of type A, and obeys

T_{i} f - s_{i} (f) T_{i} = (Q - 1) D_{i} (f) (f \in A) .

Consequently the assignments $X_{j} \mapsto X_{j} [1]$ and $T_{i} \mapsto T_{i}$ define an $R_{0}$ -algebra homomorphism from the presented affine Hecke algebra to $K ⋊ S_{n}$ . The coefficient of $[s_{i}]$ in $T_{i}$ is nonzero in $K$ .

Facts & Assumptions

Given: The universal coefficient ring $R_{0}$ , Laurent algebra $A$ , fraction field $K$ , type-A action, divided differences $D_{i}$ , and the presented algebra $H_{n}^{a f f}$ .

[F1]

The type-A weight lattice, Laurent ring, and divided differences: $A$ is a domain, $D_{i} : A \to A$ , $D_{i} (f g) = D_{i} (f) g + s_{i} (f) D_{i} (g)$ , and the adjacent transposition interchanges $X_{i}, X_{i + 1}$ .

[F2]

The rational skew group algebra for type A: $K ⋊ S_{n}$ has basis $[w]$ over $K$ and multiplication $(a [w]) (b [v]) = a w (b) [w v]$ .

[F3]

The multiplicative Bernstein presentation of the affine Hecke algebra of type A: the defining relations are commuting invertible $X_{j}$ , $(T_{i} - Q) (T_{i} + 1) = 0$ , the finite type-A braid relations, and the Bernstein cross-relations on the $X_{j}$ .

[F4]

Multivariate polynomial and Laurent rings over commutative rings, domains and fraction fields: $A$ has its Laurent monomial basis and embeds in the fraction field $K$ .

Proof

Given: The data of the Statement.

1.1givenF1F2F4

Put $c (z) : = (Q - 1) / (1 - z)$ and $d (z) : = (1 - Q z) / (1 - z) = Q - c (z)$ . In the skew group algebra, $T_{i} = c (r_{i}) [1] + d (r_{i}) [s_{i}]$ . Both $1 - r_{i}$ and $1 - Q r_{i}$ are nonzero in $A$ , because their two terms are distinct Laurent monomials over $R_{0}$ by [F4]. Hence $c (r_{i}), d (r_{i})$ are defined in $K$ and $d (r_{i}) \neq 0$ .

2.1F1F2step 1.1

For $f \in A$ , multiplication in [F2] gives $T_{i} f - s_{i} (f) T_{i} = c (r_{i}) (f - s_{i} (f)) = (Q - 1) D_{i} (f)$ . As an operator on $A$ , $T_{i} (f) = Q s_{i} (f) + (Q - 1) D_{i} (f)$ , which lies in $A$ by [F1]. Thus it preserves $A$ and satisfies the full Bernstein relation, not only its values on generators.

2.2F2step 1.1algebra

The rational functions $c, d$ satisfy $c (z) + c (z^{- 1}) = Q - 1$ , $d (z) = Q - c (z)$ , and $d (z^{- 1}) = 1 + c (z)$ by direct simplification. Expanding $(c (r_{i}) [1] + d (r_{i}) [s_{i}])^{2}$ with [F2], its $[s_{i}]$ coefficient is $d (r_{i}) (c (r_{i}) + c (r_{i}^{- 1})) = (Q - 1) d (r_{i})$ , and its $[1]$ coefficient is $c (r_{i})^{2} + d (r_{i}) d (r_{i}^{- 1}) = c (r_{i})^{2} + (Q - c (r_{i})) (1 + c (r_{i})) = Q + (Q - 1) c (r_{i})$ . Therefore $T_{i}^{2} = (Q - 1) T_{i} + Q$ , equivalent to $(T_{i} - Q) (T_{i} + 1) = 0$ .

2.3F1F2step 1.1

If $∣ i - j ∣ > 1$ , the transpositions $s_{i}, s_{j}$ commute and each fixes the coefficients $c (r_{j}), d (r_{j})$ or $c (r_{i}), d (r_{i})$ of the other operator. Expanding their product by [F2] therefore gives $T_{i} T_{j} = T_{j} T_{i}$ .

3.1F2step 2.2algebra

For adjacent indices put $s = s_{i}$ , $t = s_{i + 1}$ , $r = X_{i} / X_{i + 1}$ , $u = X_{i + 1} / X_{i + 2}$ , and abbreviate $c_{r} = c (r)$ , $c_{u} = c (u)$ , $c_{r u} = c (r u)$ and similarly for $d$ . The identities in step 2.2 give $c (r^{- 1}) = Q - 1 - c_{r}$ , $c (u^{- 1}) = Q - 1 - c_{u}$ , and $c_{r u} (Q - 1 - c_{r} - c_{u}) = - c_{r} c_{u}$ , since the left side is $- (Q - 1)^{2} / ((1 - r) (1 - u))$ . The variable permutations give $s (u) = t (r) = r u$ , $s t (r) = u$ , and $t s (u) = r$ .

4.1F2step 2.2step 3.1algebra

Expanding $T_{s} T_{t} T_{s}$ and $T_{t} T_{s} T_{t}$ by [F2], their coefficient pairs, in the order $[1], [s], [t], [s t], [t s], [s t s] = [t s t]$ , are respectively: $[1]$ : $(c_{r}^{2} c_{u} + d_{r} c_{r u} d (r^{- 1}), c_{u}^{2} c_{r} + d_{u} c_{r u} d (u^{- 1}))$ ; $[s]$ : $(d_{r} (c_{r} c_{u} + c_{r u} c (r^{- 1})), c_{u} d_{r} c_{r u})$ ; $[t]$ : $(c_{r} d_{u} c_{r u}, d_{u} (c_{r u} c (u^{- 1}) + c_{u} c_{r}))$ ; $[s t]$ : $(d_{r} d_{r u} c_{u}, c_{u} d_{r} d_{r u})$ ; $[t s]$ : $(c_{r} d_{u} d_{r u}, d_{u} d_{r u} c_{r})$ ; and $[s t s]$ : $(d_{r} d_{r u} d_{u}, d_{u} d_{r u} d_{r})$ . The last three pairs agree by commutativity of $K$ . For the $[s]$ pair, substituting $c (r^{- 1}) = Q - 1 - c_{r}$ reduces equality to $c_{r} c_{u} + c_{r u} (Q - 1 - c_{r} - c_{u}) = 0$ ; for $[t]$ the same identity gives equality after substituting $c (u^{- 1}) = Q - 1 - c_{u}$ . For $[1]$ , using $d_{r} = Q - c_{r}$ , $d (r^{- 1}) = 1 + c_{r}$ and their $u$ analogues, the difference of the entries is $(c_{r} - c_{u}) (c_{r} c_{u} + c_{r u} (Q - 1 - c_{r} - c_{u})) = 0$ . Hence the adjacent braid relation holds.

5.1F2F3step 1.1step 2.1step 2.2step 2.3step 4.1∎

The Bernstein, quadratic, distant, and adjacent braid calculations show that the operators satisfy every relation in [F3], including the commuting Laurent variables and their inverses. The universal property of the presented quotient therefore gives the asserted homomorphism. Step 1.1 established that the coefficient $d (r_{i})$ of $[s_{i}]$ is nonzero. No Choice principle or specialization is used.

Depends on

Used by

Cited to discharge well-definedness by The type-A weight lattice, Laurent ring, and divided differences.

Dependency tree · two levels

27 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Alexander A. Kirillov, Jr., Lectures on the affine Hecke algebras and Macdonald conjectures (Harvard lectures, 1994; arXiv:math/9501219v1) (standard reference, not scraped)
I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Séminaire Bourbaki 797, Astérisque 237 (1996), pp. 189–207 (standard reference, not scraped)
Eric Opdam and Maarten Solleveld, Homological algebra for affine Hecke algebras (2007) (standard reference, not scraped)