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The multiplicative Bernstein presentation of the affine Hecke algebra of type A

Definition

Let $n \geq 2$ , $R_{0} = Z [Q, Q^{- 1}]$ , and $A = R_{0} [X_{1}^{\pm 1}, \dots, X_{n}^{\pm 1}]$ as in The type-A weight lattice, Laurent ring, and divided differences. Let $F$ be the free associative unital $R_{0}$ -algebra on symbols $T_{1}, \dots, T_{n - 1}, X_{1}^{\pm 1}, \dots, X_{n}^{\pm 1}$ and let $I$ be the two-sided ideal generated by the following relations:

The $X_{j}$ commute and $X_{j} X_{j}^{- 1} = X_{j}^{- 1} X_{j} = 1$ .
$(T_{i} - Q) (T_{i} + 1) = 0$ for $1 \leq i < n$ .
The $T_{i}$ satisfy the type- $A_{n - 1}$ braid relations: $T_{i} T_{i + 1} T_{i} = T_{i + 1} T_{i} T_{i + 1}$ for $1 \leq i < n - 1$ , and $T_{i} T_{j} = T_{j} T_{i}$ when $∣ i - j ∣ > 1$ .
The Bernstein cross-relations on the Laurent generators are

T_{i} X_{j} = s_{i} (X_{j}) T_{i} + (Q - 1) D_{i} (X_{j}) (1 \leq i < n, 1 \leq j \leq n) .

Define

H_{n}^{a f f} : = F / I .

The coefficient map $A \to H_{n}^{a f f}$ sends each Laurent monomial to the corresponding word in the $X_{j}$ ; its injectivity and the resulting $A$ -module basis are the conclusion of Bernstein PBW basis for the affine Hecke algebra of type A ↗, not an assumption in this definition. The normalization is multiplicative: $Q$ is a unit, $T_{i}^{2} = (Q - 1) T_{i} + Q$ , and $T_{i}$ is invertible. It matches the finite Hecke normalization of The reversal anti-involution, generator invertibility, the bar operator and the multiplicative normalization under $Q = v^{2}$ and $T_{i} = v U_{i}$ , where $(U_{i} - v) (U_{i} + v^{- 1}) = 0$ .

Facts & Assumptions

Given: $n \geq 2$ , the Laurent algebra $A$ , its type- $A$ action and divided differences, and the free associative algebra and quotient convention.

[F1]

The type-A weight lattice, Laurent ring, and divided differences: $A$ is generated as a commutative $R_{0}$ -algebra by $X_{j}^{\pm 1}$ , $D_{i} (f) \in A$ , and $D_{i} (f g) = D_{i} (f) g + s_{i} (f) D_{i} (g)$ .

[F2]

The free associative R-algebra on a set and descent of relations: the free associative algebra on a generator set exists over any commutative coefficient ring, and quotienting by the two-sided ideal of the listed relations defines the presented unital associative algebra with its universal property.

[F3]

The reversal anti-involution, generator invertibility, the bar operator and the multiplicative normalization: in the finite Coxeter Hecke algebra, if the odd-connected type-A generators share parameter $v$ , then $U_{i}$ satisfies $(U_{i} - v) (U_{i} + v^{- 1}) = 0$ , and $T_{i} = v U_{i}$ , $Q = v^{2}$ gives $(T_{i} - Q) (T_{i} + 1) = 0$ .

Proof

Given: The data of the Definition.

1.1givenF2

The coefficient ring $R_{0} = Z [Q, Q^{- 1}]$ is commutative and $Q$ is a unit. By [F2], the free associative algebra $F$ and the two-sided ideal generated by the displayed relation set exist, so $H_{n}^{a f f} = F / I$ is a unital associative $R_{0}$ -algebra with the stated generator-and-relation interpretation.

1.2F1algebra

The divided differences on the Laurent generators are $D_{i} (X_{i}) = - X_{i + 1}$ , $D_{i} (X_{i + 1}) = X_{i + 1}$ , and $D_{i} (X_{j}) = 0$ when $j \notin {i, i + 1}$ , directly from their definition and the action of $s_{i}$ . Substitution into relation 4 gives $T_{i} X_{i} - X_{i + 1} T_{i} = - (Q - 1) X_{i + 1}$ , $T_{i} X_{i + 1} - X_{i} T_{i} = (Q - 1) X_{i + 1}$ , and $T_{i} X_{j} = X_{j} T_{i}$ for the other variables.

2.1F1step 1.2algebra

If the Bernstein relation holds for $f, g \in A$ , then using it twice and [F1] gives $T_{i} (f g) - s_{i} (f g) T_{i} = (Q - 1) (D_{i} (f) g + s_{i} (f) D_{i} (g)) = (Q - 1) D_{i} (f g)$ . For invertible $f \in A$ , the Leibniz rule applied to $f f^{- 1} = 1$ gives $D_{i} (f^{- 1}) = - s_{i} (f)^{- 1} D_{i} (f) f^{- 1}$ . From $T_{i} f = s_{i} (f) T_{i} + (Q - 1) D_{i} (f)$ , multiply on the right by $f^{- 1}$ and on the left by $s_{i} (f)^{- 1}$ to obtain $T_{i} f^{- 1} - s_{i} (f)^{- 1} T_{i} = - (Q - 1) s_{i} (f)^{- 1} D_{i} (f) f^{- 1} = (Q - 1) D_{i} (f^{- 1})$ . Since every Laurent monomial is a product of the $X_{j}$ and their inverses, additivity and induction on products prove the Bernstein relation for every $f \in A$ .

2.2step 1.2algebra

The quadratic relation is equivalent to $T_{i}^{2} = (Q - 1) T_{i} + Q$ , so step 1.2 gives $T_{i} X_{i} T_{i} = X_{i + 1} T_{i}^{2} - (Q - 1) X_{i + 1} T_{i} = Q X_{i + 1}$ . Conversely, in an algebra satisfying $T_{i}^{2} = (Q - 1) T_{i} + Q$ , with $Q \in R_{0}^{\times}$ , and $T_{i} X_{i} T_{i} = Q X_{i + 1}$ , multiplying the last equality on the right by $T_{i}$ and using the quadratic relation gives $T_{i} X_{i} - X_{i + 1} T_{i} = - (Q - 1) X_{i + 1}$ ; multiplying on the left by $T_{i}$ gives $T_{i} X_{i + 1} - X_{i} T_{i} = (Q - 1) X_{i + 1}$ . Together with the stated commutation for $j \notin {i, i + 1}$ , these are exactly the generator cross-relations of part 4. Thus the Bernstein relations and the compact relation are equivalent under the declared conventions.

3.1F3algebra∎

The quadratic relation gives $T_{i} (T_{i} - (Q - 1)) = Q = (T_{i} - (Q - 1)) T_{i}$ . Since $Q$ is a unit, $T_{i}^{- 1} = Q^{- 1} (T_{i} - (Q - 1))$ , proving the invertibility claim. The finite Hecke normalization agrees with [F3]: substituting $T_{i} = v U_{i}$ and $Q = v^{2}$ into $(U_{i} - v) (U_{i} + v^{- 1}) = 0$ gives $(T_{i} - Q) (T_{i} + 1) = v^{2} (U_{i} - v) (U_{i} + v^{- 1}) = 0$ . Every finite braid word has the same number of generators on each side, so scaling every generator by $v$ multiplies both sides by the same power of $v$ and preserves the braid relations.

Remarks

The quotient presentation itself does not assume a PBW basis or faithful inclusion of $A$ . The theorem listed in justified_by proves that inclusion and the $A$ -module basis.

Depends on

Used by

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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)