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The affine Hecke algebra and its commuting action on tensor powers of the vector representation

Definition

Fix $e \geq 2$ and keep all conventions of The level-zero vector representation, its affinization and the level-one crystal energy: the Cartan datum, the space $V_{a f f} = ⨁_{m \in Z} Q (q) u_{m}$ with $u_{m} = z^{a} v_{j}$ ( $m = j - a e$ , $j \in {1, \dots, e}$ ), the operators $K_{h}, E_{i}, F_{i}$ , the classical weights $λ_{j} = Λ_{j} - Λ_{j - 1}$ , and the fixed coproduct $Δ (K_{h}) = K_{h} \otimes K_{h}$ , $Δ (E_{i}) = E_{i} \otimes K_{i}^{- 1} + 1 \otimes E_{i}$ , $Δ (F_{i}) = F_{i} \otimes 1 + K_{i} \otimes F_{i}$ of The Drinfeld–Jimbo formulas define a Hopf algebra. Put $Q : = q^{2}$ and $c : = Q - 1$ , and let $Y_{j}$ denote the operator "multiply the $j$ -th tensor factor by $z$ ", so that $Y_{j}$ is invertible with $Y_{j} u_{m} = u_{m - e}$ in the $j$ -th slot and different $Y_{j}$ commute.

The affine Hecke algebra ${\tilde{H}}_{N}$ . For $N \geq 1$ , ${\tilde{H}}_{N}$ is the unital $Q (q)$ -algebra with generators $S_{1}, \dots, S_{N - 1}$ and invertible $Y_{1}^{\pm 1}, \dots, Y_{N}^{\pm 1}$ subject to

$S_{i}^{2} = (Q - 1) S_{i} + Q, S_{i} S_{i + 1} S_{i} = S_{i + 1} S_{i} S_{i + 1}, S_{i} S_{j} = S_{j} S_{i} (∣ i - j ∣ \geq 2),$

$Y_{i} Y_{j} = Y_{j} Y_{i}, S_{i} Y_{j} = Y_{j} S_{i} (j \notin {i, i + 1}), S_{i} Y_{i} S_{i} = Q Y_{i + 1} .$

The element $t_{i} : = - q^{- 1} S_{i}$ then satisfies $(t_{i} - q^{- 1}) (t_{i} + q) = 0$ ; for a reduced word $w = s_{i_{1}} \dots s_{i_{k}} \in S_{N}$ put $S_{w} : = S_{i_{1}} \dots S_{i_{k}}$ and $A_{N} : = \sum_{w \in S_{N}} S_{w}$ (well defined by the braid relations).

The two-vector operator and its extension. On residue tensors $v_{j} \otimes v_{k}$ ( $j, k \in {1, \dots, e}$ ) set

$S (v_{j} \otimes v_{k}) = {\begin{cases} - q v_{k} \otimes v_{j}, & j < k, \\ - v_{j} \otimes v_{j}, & j = k, \\ - q v_{k} \otimes v_{j} + (Q - 1) v_{j} \otimes v_{k}, & j > k . \end{cases}$

For a Laurent monomial $f = Y_{1}^{p} Y_{2}^{q}$ define, with $s (f) : = Y_{2}^{p} Y_{1}^{q}$ the swap of the two loop variables,

$S (f w) : = s (f) S (w) + (1 - Q) Y_{2} \frac{f - s (f)}{Y_{1} - Y_{2}} w (w \in V \otimes V),$

and extend linearly; the quotient $(f - s (f)) / (Y_{1} - Y_{2})$ is a Laurent polynomial, so $S$ is a well-defined linear endomorphism of $V_{a f f} \otimes V_{a f f}$ . On $V_{a f f}^{\otimes N}$ the generator $S_{i}$ acts by this operator on the tensor factors $i, i + 1$ and $Y_{j}$ acts on the $j$ -th factor as above; this defines the affine Hecke action. The whole action commutes with the derived quantum group $U_{q}^{'} (g)$ (the generators $E_{i}, F_{i}$ and $K_{h}$ with $h$ in the span of the simple coroots). Each $S_{i}$ , and therefore $A_{N}$ , also preserves total loop degree and commutes with the full affine quantum-group action. The individual loops instead satisfy $K_{d} Y_{j} K_{d}^{- 1} = q Y_{j}$ . Products are composed as left operators; right-action notation reverses their order using the anti-involution fixing the generators.

Facts & Assumptions

Given: The Laurent tensor space and its vector action from The level-zero vector representation, its affinization and the level-one crystal energy, the displayed operators $S_{i}, Y_{j}$ , and the fixed coproduct from The Drinfeld–Jimbo formulas define a Hopf algebra.

[F1]

The vector basis has $u_{m} = z^{a} v_{j}$ with $m = j - a e$ , $E_{i} u_{m} = δ_{m \equiv i + 1} u_{m - 1}$ , $F_{i} u_{m} = δ_{m \equiv i} u_{m + 1}$ , and $K_{i} v_{j} = q^{δ_{i, j} - δ_{i, j - 1}} v_{j}$ . In particular $E_{0} = z E_{0}^{f i n}$ , $F_{0} = z^{- 1} F_{0}^{f i n}$ , $K_{d} u_{m} = q^{a} u_{m}$ , and $w t (u_{m - e}) = w t (u_{m}) + δ$ (The level-zero vector representation, its affinization and the level-one crystal energy).

[F2]

The coproduct is $Δ (K_{h}) = K_{h} \otimes K_{h}$ , $Δ (E_{i}) = E_{i} \otimes K_{i}^{- 1} + 1 \otimes E_{i}$ , $Δ (F_{i}) = F_{i} \otimes 1 + K_{i} \otimes F_{i}$ , and is coassociative (The Drinfeld–Jimbo formulas define a Hopf algebra).

Verification

technique · establish the finite residue identities and the divided-difference identities, combine them for the full Laurent Hecke action, and verify coproduct commutation in an ordered residue window for each color

1.1givenalgebra

Write $x = Y_{1}, y = Y_{2}$ , let $s$ swap them, and set $D (f) = y (f - s (f)) / (x - y)$ . For an integer $l > 0$ , $(x^{l} - y^{l}) / (x - y) = \sum_{r = 0}^{l - 1} x^{l - 1 - r} y^{r}$ ; for $l < 0$ it is $- \sum_{r = 1}^{- l} x^{- r} y^{r + l - 1}$ , obtained by factoring $x^{- l} y^{- l}$ from the positive formula, and for $l = 0$ it is zero. Factoring a common Laurent monomial proves that $D$ preserves the Laurent ring. The identity $D (f g) = D (f) g + s (f) D (g)$ follows by subtracting $s (f) s (g)$ from $f g$ . With $d = y / (x - y)$ , $s (d) = - x / (x - y)$ and $d + s (d) = - 1$ , expansion of $D = d (1 - s)$ gives $D^{2} = - D$ and $s D + D s = 1 - s$ . These rational computations are identities on Laurent polynomials because the finite sums show that all the operators preserve that ring.

1.2givenalgebra

Let $T$ be the displayed residue matrix. For $a = v_{j} \otimes v_{k}$ , $b = v_{k} \otimes v_{j}$ , $j < k$ , one has $T^{2} a = Q a - q c b$ and $T^{2} b = - q c a + (Q + c^{2}) b$ , while $T^{2} d_{j} = d_{j}$ on $d_{j} = v_{j} \otimes v_{j}$ . Hence $T^{2} = c T + Q$ . The finite braid is also an exact finite residue identity: abbreviate a triple by its order pattern $i j k$ , where $1 < 2 < 3$ , and let $L = T_{1} T_{2} T_{1}$ , $R = T_{2} T_{1} T_{2}$ . Applying the three-case matrix gives the following common values: $L (111) = R (111) = - 111$ ; $L (112) = R (112) = - Q 211$ ; $L (121) = R (121) = - Q 121 + q c 211$ ; $L (211) = R (211) = - Q 112 + q c 121 + c 211$ ; $L (122) = R (122) = - Q 221$ ; $L (212) = R (212) = - Q 212 + q c 221$ ; $L (221) = R (221) = - Q 122 + q c 212 + c 221$ ; $L (123) = R (123) = - q^{3} 321$ ; $L (132) = R (132) = - q^{3} 231 + Q c 321$ ; $L (213) = R (213) = - q^{3} 312 + Q c 321$ ; $L (231) = R (231) = - q^{3} 132 + Q c (312 + 231) - q c^{2} 321$ ; $L (312) = R (312) = - q^{3} 213 + Q c (312 + 231) - q c^{2} 321$ ; $L (321) = R (321) = - q^{3} 123 + Q c (213 + 132) - q c^{2} (312 + 231) + (c^{3} + Q c) 321$ . These thirteen patterns exhaust all triples of residues, including repeated residues for $e = 2$ , proving the finite braid for every $e$ .

1.3F1F2givenalgebra

For a nonzero color $i$ , let $A = v_{i}$ be its top vector and $B = v_{i + 1}$ its bottom; $K_{i}$ has eigenvalues $q, q^{- 1}$ there and $1$ on every other residue $O$ . On the four pairs $A A, A B, B A, B B$ , the coproduct maps are $Δ (E_{i}) : (0, A A, q^{- 1} A A, q A B + B A)$ and $Δ (F_{i}) : (B A + q A B, B B, q^{- 1} B B, 0)$ respectively. Applying $T$ before or after them gives respectively $T Δ (E_{i}) = Δ (E_{i}) T : (0, - A A, - q^{- 1} A A, - q A B - B A)$ and $T Δ (F_{i}) = Δ (F_{i}) T : (- B A - q A B, - B B, - q^{- 1} B B, 0)$ . For spectator pairs, $B O \mapsto A O$ , $O B \mapsto O A$ under $E_{i}$ , and $A O \mapsto B O$ , $O A \mapsto O B$ under $F_{i}$ , with other pairs killed. Since $A, B$ are consecutive residues, every spectator is below both or above both, so the order comparison and the coefficients in $T$ are unchanged by these maps. This proves residue commutation in every case. $T$ also preserves each total derived toral weight because it only permutes residues.

2.1step 1.1step 1.2algebra

On the Laurent tensor space the operator is $S = T s - c D$ , with $T$ acting only on residues. It commutes with $s, D$ on coefficients. Thus $S^{2} = T^{2} - c T (s D + D s) + c^{2} D^{2} = c T + Q - c T (1 - s) - c^{2} D = c S + Q$ by steps 1.1 and 1.2. This proves the quadratic relation on all loop powers, rather than only on residue tensors. Off-adjacent operators commute because their residue matrices and divided differences involve disjoint positions.

2.2step 1.1step 1.3F1F2algebra

For $i \neq 0$ , $Δ (E_{i}), Δ (F_{i})$ and the derived torals act by constant residue matrices and commute with scalar Laurent multiplication and the coefficient operators $s, D$ . Together with step 1.3 and $S = T s - c D$ this proves their commutation with $S$ on every Laurent tensor. In particular the correct extension of a commutator is $[S, Δ (X)] (f w) = s (f) [S, Δ (X)] w$ , since the divided-difference terms cancel; it is not a formula with $f$ instead of $s (f)$ . More generally step 1.1 gives $S (f h) = s (f) S (h) - c D (f) h$ for every Laurent tensor $h$ , by applying the Leibniz identity to its residue coefficients.

3.1step 1.1step 1.2step 2.1algebra

For three adjacent loop variables $z_{1}, z_{2}, z_{3}$ put $b_{i j} = - c z_{j} / (z_{i} - z_{j})$ , $a = b_{12}$ , $b = b_{23}$ , and $h = b_{13}$ . Then $b_{i j} + b_{j i} = c$ and $δ_{0} : = a b - h (a + b - c) = 0$ , the latter following on multiplying by $(z_{1} - z_{2}) (z_{2} - z_{3}) (z_{1} - z_{3})$ . Write $S_{i} = (T_{i} - b_{i, i + 1}) s_{i} + b_{i, i + 1}$ . Expand $S_{1} S_{2} S_{1} - S_{2} S_{1} S_{2}$ with every coefficient on the left, using $s_{i} b_{j k} = b_{s_{i} (j), s_{i} (k)} s_{i}$ . After using $T_{i}^{2} = c T_{i} + Q$ and $T_{1} T_{2} T_{1} = T_{2} T_{1} T_{2}$ , the coefficients at $1, s_{1}, s_{2}, s_{1} s_{2}, s_{2} s_{1}, s_{1} s_{2} s_{1}$ are respectively $(a - b) δ_{0}$ , $(T_{1} - a) δ_{0}$ , $- (T_{2} - b) δ_{0}$ , $0$ , $0$ , and $- (T_{1} - T_{2}) δ_{0}$ . For example the coefficient at $1$ is $a^{2} b + h (Q + a (c - a)) - b^{2} a - h (Q + b (c - b)) = (a - b) δ_{0}$ ; the longest coefficient is $(T_{1} - a) (T_{2} - h) (T_{1} - b) - (T_{2} - b) (T_{1} - h) (T_{2} - a) = - (T_{1} - T_{2}) δ_{0}$ . Hence all coefficients vanish. Clearing the three differences gives the identity in the original Laurent tensor space, whose injection into its rational coefficient space is faithful; the operators already preserve the Laurent space by step 1.1. This proves the full braid, with no finite-loop truncation assumption.

4.1step 1.1step 2.1step 3.1algebra

The loop multiplication operators are invertible and commute. From step 1.1 and the extension formula, for $l > 0$ one has $S x^{l} = y^{l} S - c \sum_{r = 1}^{l} x^{l - r} y^{r}$ ; for $l < 0$ one has $S x^{l} = y^{l} S + c \sum_{r = 1}^{- l} x^{- r} y^{r + l}$ , and for $l = 0$ there is no correction. These identities hold on every tensor by the divided-difference Leibniz formula. In particular $S x = y S - c y$ and $S y = x S + c y$ . Therefore $S x S = y S^{2} - c y S = Q y$ by step 2.1. These are precisely the Bernstein and off-adjacent loop relations in the Definition. Finally $t = - q^{- 1} S$ satisfies $(t - q^{- 1}) (t + q) = q^{- 2} (S + 1) (S - Q) = 0$ , and equivalently $t^{2} = (q^{- 1} - q) t + 1$ .

5.1step 1.1step 4.1step 1.3step 2.2F1F2algebra

For the affine color use the ordered Laurent basis $w_{1} = z v_{e} = u_{0}$ , $w_{2} = v_{1} = u_{1}$ , through $w_{e} = v_{e - 1} = u_{e - 1}$ . For $1 \leq j < e$ , the extension formula gives $S (x (v_{e} \otimes v_{j})) = - q y (v_{j} \otimes v_{e})$ and $S (y (v_{j} \otimes v_{e})) = - q x (v_{e} \otimes v_{j}) + c y (v_{j} \otimes v_{e})$ ; also $S (x y (v_{e} \otimes v_{e})) = - x y (v_{e} \otimes v_{e})$ . On the other pairs the residue formula is unchanged. Thus $S$ on $w_{a} \otimes w_{b}$ is exactly the same three-case matrix in the order $w_{1} < \dots < w_{e}$ , and its extension remains the formula in step 2.2. On this basis $E_{0} w_{2} = w_{1}$ , $F_{0} w_{1} = w_{2}$ , with all other images zero, because $E_{0} = z E_{0}^{f i n}$ and $F_{0} = z^{- 1} F_{0}^{f i n}$ . Their toral eigenvalues are $q, q^{- 1}$ on $w_{1}, w_{2}$ and $1$ on spectators. Consequently precisely the finite commutation calculation of step 1.3 applies to color $0$ , followed by the constant-matrix Laurent argument of step 2.2. This proves all affine-color commutators on arbitrary positive and negative loop powers.

6.1step 1.1step 2.2step 5.1F1F2algebra

On two factors $S$ preserves total residue weight and total loop degree: $s$ swaps loop powers and $D$ has total degree zero, since its numerator is multiplied by $y$ and divided by $x - y$ . Hence it commutes with $K_{h} \otimes K_{h}$ for every full toral $K_{h}$ , including $K_{d}$ . Each $Y_{j}$ commutes with $E_{i}, F_{i}$ and derived torals by [F1], but increases total loop degree by one, giving $K_{d} Y_{j} K_{d}^{- 1} = q Y_{j}$ . For $N > 2$ , the iterated coproduct of $E_{i}$ is the sum of terms $1^{\otimes (r - 1)} \otimes E_{i} \otimes (K_{i}^{- 1})^{\otimes (N - r)}$ , and that of $F_{i}$ is the sum $K_{i}^{\otimes (r - 1)} \otimes F_{i} \otimes 1^{\otimes (N - r)}$ . The two terms in the adjacent slots combine into their two-factor coproduct; a term outside those slots has either $K_{i}^{\pm 1} \otimes K_{i}^{\pm 1}$ or $1 \otimes 1$ there. Steps 2.2 and 5.1 and the toral commutation therefore prove $[S_{j}, Δ^{(N)} (X)] = 0$ for every full generator $X$ . Individual $Y_{j}$ commute with the derived action for the same slotwise reason.

7.1step 2.1step 3.1step 4.1step 2.2step 5.1step 6.1∎

The quadratic, braid, distant, loop and Bernstein identities prove the displayed affine Hecke action on the entire Laurent tensor space. The finite Hecke operators $S_{i}$ commute with the full quantum-group action; their products $S_{w}$ are well defined by reduced words and their sum $A_{N}$ has the same commutation and total-degree properties. The entire affine Hecke action, including each individual loop, commutes with the derived action, with the degree covariance of step 6.1. This establishes exactly the scope and normalization claimed in the Definition.

Remarks

The loops have degree one, whereas $S_{i}$ and $A_{N}$ have degree zero. The action uses the fixed coproduct throughout; flipping to the opposite channel is a separate convention change. At $q = 1$ , $S_{i}$ is the negative flip, including its Laurent coefficients.

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Sources

Bernard Leclerc, Fock space representations (author-hosted complete 38-page treatment) (standard reference, not scraped)
Masaki Kashiwara, Tetsuji Miwa and Eugene Stern, Decomposition of q-deformed Fock spaces, arXiv:q-alg/9508006 (standard reference, not scraped)
Kashiwara-Miwa-Petersen-Yung, Perfect crystals and q-deformed Fock spaces (standard reference, not scraped)