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Cartan projection constructs the Verma contravariant form and identifies its radical

Statement

Let $λ \in P$ , let $M (λ)$ be the Verma module with its canonical $P$ -label grading and highest vector $v_{λ}$ (Quantum Verma modules $M (λ)$ , highest weight vectors, and the simple highest weight quotient $L (λ)$ ), and let $σ$ be the linear contravariant anti-involution of The contravariant form on an integrable highest weight module ( $σ (K_{h}) = K_{h}$ , $σ (E_{i}) = q_{i} F_{i} K_{i}^{- 1}$ , $σ (F_{i}) = q_{i}^{- 1} K_{i} E_{i}$ ).

(i) Existence and uniqueness. Define the $λ$ -weight functional $η ⁣ : U_{q} (g) \to Q (q)$ by $η (y) : =$ the coefficient of $v_{λ}$ in $y v_{λ}$ (so $η (y)$ is unchanged by adding an element of the Verma ideal $J (λ)$ , and $η (1) = 1$ ). Then $B (a v_{λ}, b v_{λ}) : = η (σ (a) b) (a, b \in U_{q}^{-})$ is a well-defined bilinear form on $M (λ)$ ; it is homogeneous, normalized ( $B (v_{λ}, v_{λ}) = 1$ ) and $σ$ -contravariant, $B (x u, v) = B (u, σ (x) v)$ . It is the unique such form on $M (λ)$ , in the sense of The contravariant form on an integrable highest weight module.

(ii) Radical and descent. The radical $rad B = {u : B (u, M (λ)) = 0}$ equals the sum $N (λ)$ of all proper graded submodules of $M (λ)$ ; consequently $B$ descends to a nondegenerate normalized $σ$ -contravariant form on the simple highest weight quotient $L (λ) = M (λ) / N (λ)$ .

(iii) Rank-one norms and dominant singular vectors. If $m_{i} = ⟨ λ, h_{i} ⟩ \geq 0$ (that is, $λ$ dominant), then for every $r \geq 0$ $B (F_{i}^{(r)} v_{λ}, F_{i}^{(r)} v_{λ}) = q_{i}^{r (m_{i} - r)} {(\binom{m_{i}}{r})}_{i},$ with the Gaussian binomial interpreted as $0$ for $r > m_{i}$ ; in particular each $F_{i}^{(m_{i} + 1)} v_{λ}$ lies in the radical, so $B$ factors through the Serre quotient $M (λ) / \sum_{i} U_{q} (g) F_{i}^{(m_{i} + 1)} v_{λ}$ . (The same computation proves the norm formula for every $λ$ and all $r$ for which the right-hand side is read with $m_{i} = ⟨ λ, h_{i} ⟩$ .)

(iv) Ordinary simplicity. Let $V$ be an integrable cyclic graded quotient of $M (λ)$ with support in $λ - Q_{+}$ on which the descended form is nondegenerate. Then $V$ is simple as an ordinary module (not only as a graded module).

(v) Integrality of the lattice values. If $λ$ is integral and $V_{A} : = U_{A} v_{λ} = U_{A}^{-} v_{λ} \subseteq L (λ)$ is the highest lattice of the Lusztig form (The Lusztig divided-power integral form $U_{A}$ ), then $B (V_{A}, V_{A}) \subseteq A = Z [q, q^{- 1}]$ . No perfectness of $B$ over $A$ is asserted.

Facts & Assumptions

Given: $λ \in P$ , the Verma module $M (λ) = U_{q}^{-} v_{λ}$ with label grading, the anti-involution $σ$ , and the functional $η (y) =$ the coefficient of $v_{λ}$ in $y v_{λ}$ .

[F1]

$M (λ) = U_{q} v_{λ}$ , $M (λ)_{λ - ν} = U_{q}^{-} [- ν] v_{λ}$ , $M (λ)_{λ} = Q (q) v_{λ}$ ; the map $U_{q}^{-} \to M (λ)$ , $a \mapsto a v_{λ}$ , is a linear isomorphism; the Verma ideal is $J (λ) = U_{q} (g) ⟨ E_{i}, K_{h} - q^{⟨ λ, h ⟩} ⟩$ , and $y v_{λ} = 0$ if and only if $y \in J (λ)$ (Quantum Verma modules $M (λ)$ , highest weight vectors, and the simple highest weight quotient $L (λ)$ ).

[F2]

$σ$ is an involutive $Q (q)$ -linear anti-automorphism with $σ (K_{h}) = K_{h}$ , $σ (E_{i}) = q_{i} F_{i} K_{i}^{- 1}$ , $σ (F_{i}) = q_{i}^{- 1} K_{i} E_{i}$ ; it preserves the root grading up to sign ( $σ$ multiplies root degrees by $- 1$ ) (The contravariant form on an integrable highest weight module, The Chevalley involution, bar involution, and contravariant anti-involution preserve the Drinfeld–Jimbo ideal).

[F3]

$η$ is $Q (q)$ -linear and factors through $J (λ)$ ; it satisfies $η (K_{h} y) = q^{⟨ λ, h ⟩} η (y)$ for all $y$ , and $η (F_{i} \cdot U_{q} (g)) = 0$ : indeed $F_{i} y v_{λ}$ has label $μ - α_{i}$ when $y v_{λ}$ has label $μ \leq λ$ , which can equal $λ$ only if some weight of $M (λ)$ equals $λ + α_{i}$ , and all weights of $M (λ)$ lie in $λ - Q_{+}$ (Quantum Verma modules $M (λ)$ , highest weight vectors, and the simple highest weight quotient $L (λ)$ ).

[F4]

Triangular decomposition: $U_{q} (g)$ is the direct sum over $α, γ \in Q_{+}$ of the subspaces $U_{q}^{-} [- γ] U_{q}^{0} U_{q}^{+} [α]$ , and $U_{q} (g) = U_{q}^{-} U_{q}^{0} U_{q}^{+}$ as a vector space; for $y \in U_{q} (g)$ , compute $y v_{λ}$ from its normal products by killing positive-degree positive factors and evaluating the toral factors at the highest character, as in [F1] (Triangular decomposition of a quantized enveloping algebra).

[F5]

In this proof, for every integer $n$ write $[n]_{i} : = (q_{i}^{n} - q_{i}^{- n}) / (q_{i} - q_{i}^{- 1})$ , extending the nonnegative convention and giving $[- n]_{i} = - [n]_{i}$ . Rank-one reordering: for every $r \geq 1$ , $E_{i} F_{i}^{(r)} = F_{i}^{(r)} E_{i} + F_{i}^{(r - 1)} T_{i, 1 - r, 1}$ and $T_{i, 1 - r, 1} v_{λ} = [m_{i} - r + 1]_{i} v_{λ}$ with $m_{i} = ⟨ λ, h_{i} ⟩$ , while $K_{i} F_{i}^{(r - 1)} v_{λ} = q_{i}^{m_{i} - 2 (r - 1)} F_{i}^{(r - 1)} v_{λ}$ (Divided-power multiplication, toral binomials and the $E_{i}^{(a)} F_{i}^{(b)}$ reordering identity (iii), Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ ).

[F6]

Divided-power multiplication $F_{i}^{(a)} F_{i}^{(b)} = {(\binom{a + b}{a})}_{i} F_{i}^{(a + b)}$ and ${(\binom{m}{r})}_{i} = 0$ for $r > m$ , with all Gaussian coefficients integer Laurent polynomials in $q_{i}$ (Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ , The quantum Pascal recurrences, the Gauss product formula and Gaussian integrality).

Verification

technique · build the form from the $\lambda$-weight functional, verify contravariance on the three families of generators using the weight bound and the Verma ideal, prove uniqueness for every highest label and obtain symmetry by transposition, identify the radical with the sum of proper graded submodules by a pairing argument, and compute the rank-one norms by the divided-power recursion

1.1givenF1F2F4algebra

The functional $η$ and the formula. Since $a \mapsto a v_{λ}$ is a linear isomorphism $U_{q}^{-} \to M (λ)$ by [F1], the assignment $B (a v_{λ}, b v_{λ}) : = η (σ (a) b)$ is a well-defined bilinear form. It is homogeneous: if $a \in U_{q}^{-} [- ν]$ , $b \in U_{q}^{-} [- μ]$ then $σ (a) b$ has root degree $ν - μ$ , and $η$ vanishes on elements of nonzero root degree (the coefficient of the degree-zero vector $v_{λ}$ in $y v_{λ}$ vanishes when $y$ has nonzero degree, by the grading of $M (λ)$ ); hence $B$ vanishes unless $ν = μ$ . It is normalized: $B (v_{λ}, v_{λ}) = η (σ (1) 1) = η (1) = 1$ .

2.1step 1.1F1F2F3givenalgebra

Contravariance on generators. (a) For $x = F_{j}$ : $B (F_{j} a v_{λ}, b v_{λ}) = η (σ (a) σ (F_{j}) b) = B (a v_{λ}, σ (F_{j}) b v_{λ})$ is a tautology of associativity. (b) For $x = K_{h}$ : $K_{h} a v_{λ} = q^{⟨ λ - ν, h ⟩} a v_{λ}$ if $a \in U_{q}^{-} [- ν]$ , so $B (K_{h} a v_{λ}, b v_{λ}) = q^{⟨ λ - ν, h ⟩} B (a v_{λ}, b v_{λ})$ while $B (a v_{λ}, σ (K_{h}) b v_{λ}) = B (a v_{λ}, K_{h} b v_{λ})$ and $K_{h} b v_{λ} = q^{⟨ λ - μ, h ⟩} b v_{λ}$ ; both quantities vanish unless the labels match ( $ν = μ$ , forced by the homogeneity of $B$ ), in which case the two scalars agree. (c) For $x = E_{i}$ : write $E_{i} a = a E_{i} + w$ with $w = (K_{i} Q - K_{i}^{- 1} R) / (q_{i} - q_{i}^{- 1})$ , $Q = e_{i}^{''} (a)$ , $R = e_{i}^{'} (a) \in U_{q}^{-}$ (Kashiwara's skew derivations $e_{i}^{'}$ and $e_{i}^{''}$ on the negative half), so $E_{i} a v_{λ} = w v_{λ}$ and the coordinate of $E_{i} a v_{λ}$ is the normal form $n f (w) \in U_{q}^{-}$ of $w$ . For every $b \in U_{q}^{-}$ one has $E_{i} a - n f (w) \in J (λ)$ , and $σ (J (λ)) U_{q}^{-} \subseteq \ker η$ : indeed $J (λ) = U_{q} (g) ⟨ E_{i}, K_{h} - q^{⟨ λ, h ⟩} ⟩$ , so $σ (J (λ)) = ⟨ σ (E_{i}), σ (K_{h}) - q^{⟨ λ, h ⟩} ⟩ U_{q} (g)$ , and $η$ kills $(K_{h} - q^{⟨ λ, h ⟩}) U_{q} (g)$ by $η (K_{h} y) = q^{⟨ λ, h ⟩} η (y)$ of [F3], while it kills $σ (E_{i}) U_{q} (g) U_{q}^{-} = σ (E_{i}) U_{q} (g) = q_{i} F_{i} K_{i}^{- 1} U_{q} (g)$ by $η (F_{i} U_{q} (g)) = 0$ of [F3]. Hence $η (σ (n f (w)) b) = η (σ (E_{i} a) b) = η (σ (a) σ (E_{i}) b)$ , and also $η (σ (w) b) = η (σ (a) σ (E_{i}) b)$ because $σ (w) = σ (a) σ (E_{i}) - σ (E_{i}) σ (a)$ and $η$ kills the latter left factor. Therefore $B (E_{i} a v_{λ}, b v_{λ}) = η (σ (w) b) = η (σ (a) σ (E_{i}) b) = B (a v_{λ}, σ (E_{i}) b v_{λ})$ . Since the set of $x$ for which $B (x u, v) = B (u, σ (x) v)$ holds for all $u, v$ is a subalgebra of $U_{q} (g)$ (if it holds for $x$ and $y$ , then $B (x y u, v) = B (y u, σ (x) v) = B (u, σ (y) σ (x) v) = B (u, σ (x y) v)$ ) and contains the generators by (a)-(c), the form is $σ$ -contravariant.

3.1step 2.1F1F2algebra

Uniqueness and symmetry for every $λ \in P$ . Let $C, C^{'}$ be normalized $σ$ -contravariant bilinear forms on $M (λ)$ and set $D = C - C^{'}$ . For every nonempty word $w$ in the $F_{i}$ , $σ (w) v_{λ} = 0$ , because each $σ (F_{i}) = q_{i}^{- 1} K_{i} E_{i}$ kills the highest vector. Contravariance and $σ^{2} = i d$ give $D (v_{λ}, w v_{λ}) = D (σ (w) v_{λ}, v_{λ}) = 0$ ; normalization gives the same for the empty word. Since negative words span $M (λ)$ , $D (v_{λ}, M (λ)) = 0$ , and then $D (a v_{λ}, u) = D (v_{λ}, σ (a) u) = 0$ for every $a \in U_{q}^{-}$ and $u \in M (λ)$ . This proves uniqueness without dominance, symmetry or homogeneity assumptions. Now $B^{t} (u, v) : = B (v, u)$ is normalized and $σ$ -contravariant, since $B^{t} (x u, v) = B (v, x u) = B (σ (x) v, u) = B^{t} (u, σ (x) v)$ by step 2.1. Uniqueness gives $B^{t} = B$ , so $B$ is symmetric.

4.1step 1.1step 2.1step 3.1F1algebra

The radical equals $N (λ)$ . The radical of a contravariant form is a submodule ( $B (x u, v) = 0$ for $u \in rad B$ and all $v$ implies $B (u, σ (x) v) = 0$ ), it is homogeneous by step 1.1, and it is proper because $B (v_{λ}, v_{λ}) = 1$ . Hence $rad B$ is a proper graded submodule and $rad B \subseteq N (λ)$ . Conversely let $n \in N (λ)$ have label $λ - ν$ and let $b \in U_{q}^{-} [- ν]$ be arbitrary. By symmetry and contravariance, $B (n, b v_{λ}) = B (b v_{λ}, n) = B (v_{λ}, σ (b) n)$ ; now $σ (b) n \in U_{q} (g) N (λ) \subseteq N (λ)$ and $σ (b) n$ has label $λ$ because $b$ has label $ν$ and $n$ has label $λ - ν$ , so $σ (b) n \in N (λ)_{λ} = N (λ) \cap M (λ)_{λ} = 0$ because $N (λ)$ is proper graded and $M (λ)_{λ} = Q (q) v_{λ}$ with $v_{λ} \notin N (λ)$ . Hence $B (n, b v_{λ}) = 0$ ; since the $b v_{λ}$ with $b \in U_{q}^{-} [- ν]$ span $M (λ)_{λ - ν}$ and $B (n, \cdot)$ vanishes on all other graded pieces by homogeneity, $n \in rad B$ . Thus $N (λ) \subseteq rad B$ and the two agree. Consequently the induced form on $L (λ) = M (λ) / N (λ)$ is well defined, normalized, contravariant and nondegenerate (if $\overset{ˉ}{u}$ pairs to zero with all of $L (λ)$ then $u$ pairs to zero with all of $M (λ)$ , so $u \in rad B = N (λ)$ , i.e. $\overset{ˉ}{u} = 0$ ).

5.1step 2.1step 4.1F5F6algebra

Rank-one norms. Write $N_{r} : = B (F_{i}^{(r)} v_{λ}, F_{i}^{(r)} v_{λ})$ and $m = m_{i}$ . For $r \geq 1$ use contravariance with $x = F_{i}$ : since $F_{i} F_{i}^{(r - 1)} = [r]_{i} F_{i}^{(r)}$ [F6], $N_{r} = [r]_{i}^{- 1} B (F_{i} F_{i}^{(r - 1)} v_{λ}, F_{i}^{(r)} v_{λ}) = [r]_{i}^{- 1} B (F_{i}^{(r - 1)} v_{λ}, σ (F_{i}) F_{i}^{(r)} v_{λ}) = [r]_{i}^{- 1} q_{i}^{- 1} B (F_{i}^{(r - 1)} v_{λ}, K_{i} E_{i} F_{i}^{(r)} v_{λ})$ . By [F5], $E_{i} F_{i}^{(r)} v_{λ} = F_{i}^{(r - 1)} T_{i, 1 - r, 1} v_{λ} = F_{i}^{(r - 1)} [m - r + 1]_{i} v_{λ}$ , so $K_{i} E_{i} F_{i}^{(r)} v_{λ} = [m - r + 1]_{i} q_{i}^{m - 2 (r - 1)} F_{i}^{(r - 1)} v_{λ}$ and $N_{r} = [r]_{i}^{- 1} q_{i}^{m - 2 r + 1} [m - r + 1]_{i} N_{r - 1}$ . With $N_{0} = 1$ , induction gives $N_{r} = \prod_{k = 1}^{r} q_{i}^{m - 2 k + 1} \frac{[m - k + 1]_{i}}{[k]_{i}} = q_{i}^{r m - r^{2}} \frac{[m]_{i} \dots [m - r + 1]_{i}}{[1]_{i} \dots [r]_{i}} = q_{i}^{r (m - r)} {(\binom{m}{r})}_{i}$ , since $\sum_{k = 1}^{r} (m - 2 k + 1) = r m - r^{2}$ ; when $m \geq 0$ and $r > m$ the factor with $k = m + 1$ is $[0]_{i} = 0$ , so the product vanishes, and ${(\binom{m}{r})}_{i} = 0$ by [F6]. For $r = m_{i} + 1$ this gives $N_{m_{i} + 1} = 0$ ; moreover for any word $w$ of degree $- (m_{i} + 1) α_{i}$ one has $w = c F_{i}^{(m_{i} + 1)}$ because $U_{q}^{-} [- (m_{i} + 1) α_{i}]$ is spanned by the single word $F_{i}^{m_{i} + 1}$ , so $B (F_{i}^{(m_{i} + 1)} v_{λ}, w v_{λ}) = 0$ for all $w$ , i.e. $F_{i}^{(m_{i} + 1)} v_{λ} \in rad B = N (λ)$ by step 4.1.

6.1step 4.1step 5.1F1F2algebra

Ordinary simplicity. Let $V$ be the quotient of the statement and $0 \neq W \subseteq V$ an ordinary submodule. Choose $0 \neq x \in W$ and write $x = \sum_{j} x_{j}$ with nonzero homogeneous components of distinct labels $μ_{j} \in λ - Q_{+}$ . Choose a minimal label $μ_{j}$ in this finite support. Homogeneity and nondegeneracy give a homogeneous $b \in U_{q}^{-} [- (λ - μ_{j})]$ with $B (x_{j}, b {\overset{ˉ}{v}}_{λ}) \neq 0$ . The vector $y : = σ (b) x$ belongs to $W$ . For $k \neq j$ , $σ (b) x_{k}$ would have label $λ + μ_{k} - μ_{j}$ , which cannot lie in $λ - Q_{+}$ : membership would imply $μ_{k} < μ_{j}$ , contradicting minimality. Thus $y = σ (b) x_{j} \in V_{λ} = Q (q) {\overset{ˉ}{v}}_{λ}$ , and $B (y, {\overset{ˉ}{v}}_{λ}) = B (x_{j}, b {\overset{ˉ}{v}}_{λ}) \neq 0$ . Hence $W$ contains ${\overset{ˉ}{v}}_{λ}$ and equals its cyclic module $V$ . This proves ordinary simplicity without taking homogeneous components inside $W$ .

7.1step 1.1step 4.1F5F6algebra∎

Integrality. Let $λ$ be integral and $a, b \in U_{A}^{-}$ . By the stability computation of The bar involution and the contravariant anti-involution preserve $U_{A}$ (applied to $σ$ in place of the two involutions named there: $σ (E_{i}^{(r)}) = q_{i}^{r^{2}} F_{i}^{(r)} K_{i}^{- r}$ and $σ (F_{i}^{(r)}) = q_{i}^{- r^{2}} K_{i}^{r} E_{i}^{(r)}$ lie in $U_{A}$ , and $σ (K_{h}) = K_{h}$ , $σ (T_{i, c, t}) = T_{i, c, t}$ ), one has $σ (a) b \in U_{A}$ . Write $σ (a) b$ in the triangular spanning form $σ (a) b = \sum_{t} b_{t} u_{t} c_{t}$ with $b_{t} \in U_{A}^{-}$ , $u_{t} \in U_{A}^{0}$ , $c_{t} \in U_{A}^{+}$ (Divided-power multiplication, toral binomials and the $E_{i}^{(a)} F_{i}^{(b)}$ reordering identity (iv)). Then $σ (a) b v_{λ} = \sum_{t} ϵ_{t} χ_{λ} (u_{t}) b_{t} v_{λ}$ with $ϵ_{t}$ the augmentation of $c_{t}$ , so $η (σ (a) b) = \sum_{t} ϵ_{t} χ_{λ} (u_{t}) ε (b_{t})$ with $ε$ the augmentation of $U_{A}^{-}$ : each of these three factors lies in $A$ , because $ε (U_{A}^{\pm}) \subseteq A$ , $χ_{λ} (K_{h}) = q^{⟨ λ, h ⟩} \in A$ and $χ_{λ} (T_{i, c, t}) = {[⟨ λ, h_{i} ⟩ + c c h o o s e t]}_{i}$ , a Gaussian product with integer upper entry $n$ . For $n \geq 0$ it is Laurent integral by [F6]; for $n = - a < 0$ , $[n - s + 1]_{i} = - [a + s - 1]_{i}$ gives ${(\binom{- a}{t})}_{i} = (- 1)^{t} {(\binom{a + t - 1}{t})}_{i} \in A$ . Hence $B (a v_{λ}, b v_{λ}) = η (σ (a) b) \in A$ , which is (v). No perfectness of $B$ over $A$ and no $A$ -valuedness of the negative-half Kashiwara form is claimed. This completes the construction and the proof.

Depends on

Used by

Cited to discharge well-definedness by The contravariant form on an integrable highest weight module.

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Sources

Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390 (standard reference, not scraped)
George Lusztig, Introduction to Quantum Groups (Progress in Mathematics 110, Birkhauser, 1993; second printing 1994; author-hosted scan) (standard reference, not scraped)
Richard Borcherds, Mark Haiman, Theo Johnson-Freyd, Nicolai Reshetikhin and Vera Serganova, Berkeley Lectures on Lie Groups and Quantum Groups (book-length lecture notes, last updated 18 January 2024) (standard reference, not scraped)