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Lusztig's canonical basis and Kashiwara's lower global basis agree in symmetric finite ADE type

Statement

Assume symmetric finite ADE type, so $d_{i} = 1$ for all $i$ . Transport Lusztig's parameter by $v = q^{- 1}$ , the abstract half generators by $θ_{i}^{(n)} \mapsto F_{i}^{(n)}$ , and the full torus by $K_{h}^{L} \mapsto K_{- h}$ , as in The Lusztig–Kashiwara form, parameter and coproduct dictionary. Under this exact dictionary the divided-power integral forms and the bar involutions agree, and the half forms agree through the root-color factor $B_{L} = C_{ν} B_{K}$ , not by an unqualified identity. Then Lusztig's canonical lower basis equals the Kashiwara lower global basis of $U^{-}$ , and Lusztig's canonical highest-module quotient bases equal the corresponding lower global bases of The lower global basis of $U_{q}^{-}$ and of a highest weight module for every dominant label $λ$ in the explicit regular realization. The signed canonical set equals $\pm G_{\infty}$ , and the positive canonical choice is matched by its actual $T_{i, n}$ top-string recursion. No equality of upper and lower bases, and no unitriangular word-basis characterization, is asserted.

Facts & Assumptions

Given: The parameter, coproduct and form dictionary of The Lusztig–Kashiwara form, parameter and coproduct dictionary with $B_{L} = C_{ν} B_{K}$ , $C_{ν} = \prod_{i} (1 - q_{i}^{2})^{- ν_{i}}$ ; the lower global bases and their norm description (Bar invariance, congruence and uniqueness of the lower global basis, Integral residues of the negative half: freeness, the crystal basis and the form characterization, Crystal lattice polarizations are perfect at $q = 0$ ); the adapted string ideals and the exact top-string transition (The highest weight and negative half integral crystal triples are balanced); the based quotient (The based quotient: projection of the lower global basis to a highest weight module); the auxiliary regular datum (Finite-support regularization of quantum label-weight modules).

[F1]

Dictionary (The Lusztig–Kashiwara form, parameter and coproduct dictionary): the semilinear source-to-library isomorphism $φ$ carries source relations, coproduct $Δ_{L}$ and divided powers to the library presentation with $φ (θ_{i}^{(n)}) = F_{i}^{(n)}$ and $φ (K_{h}^{L}) = K_{- h}$ ; on homogeneous degree $ν$ , $B_{L} = C_{ν} B_{K}$ with $C_{ν}$ a unit of the form $\prod_{i} (1 - q_{i}^{2})^{- ν_{i}} \equiv 1 m o d q$ , and $\pm 1$ sign conventions are fixed by the generator values.

[F2]

Norm and integral criteria (Crystal lattice polarizations are perfect at $q = 0$ , Integral residues of the negative half: freeness, the crystal basis and the form characterization): $L (\infty) = {u \in U_{q}^{-} : B_{K} (u, u) \in A_{0}}$ (maximality), the pairing takes values on $L_{Z} = U_{A}^{-} \cap L (\infty)$ in $A_{Z} = Z [q, (1 - q^{2 n})^{- 1} : n \geq 1]$ , and its residue pairing is $Z$ -valued with orthonormal crystal classes in $B (\infty)$ , and the norm- $1$ elements of $L_{Z} / q L_{Z}$ are exactly $B (\infty) \cup (- B (\infty))$ ; the balanced intersection is $⨁_{b} Z G_{\infty} (b)$ with $B_{K} (G_{\infty} (b), G_{\infty} (b^{'})) \equiv δ_{b b^{'}}$ modulo $q$ .

[F3]

String partition and top-string transition (The highest weight and negative half integral crystal triples are balanced, The string-ideal triples are balanced: the induction step (Kashiwara 7.4.1 / JKK 10.4)): the adapted bases of the string ideals $I_{(i, n)}^{A}$ partition $G_{\infty} (B (\infty))$ by exact string position, and for ${\tilde{e}}_{i} b = 0$ one has $G_{\infty} ({\tilde{f}}_{i}^{n} b) - F_{i}^{(n)} G_{\infty} (b) \in I_{(i, n + 1)}^{A}$ , an exact congruence modulo the strictly higher string ideal with coefficient $1$ .

[F4]

Based quotient of the half (The based quotient: projection of the lower global basis to a highest weight module): $π_{λ}$ is a based quotient; $G_{\infty} (b)$ maps to $0$ precisely for the killed crystal classes and otherwise to $G_{λ} (π_{λ} (b))$ , bijectively onto the integral highest-module basis.

[F5]

Auxiliary transport (Finite-support regularization of quantum label-weight modules): the source $Y$ -regularity hypotheses are discharged in the explicit primitive-coroot regular datum, and characters and labels are transported back to the original lattice; the highest weight eigenvalue $v^{⟨ h, λ ⟩} = q^{- ⟨ h, λ ⟩}$ agrees with the action of $K_{- h}$ .

Proof

technique · direct

1.1F1

Form and norm transport. For $x \in U_{A}^{-} [- ν]$ one has $B_{L} (x, x) = C_{ν} B_{K} (x, x)$ by [F1], with $C_{ν}$ and $C_{ν}^{- 1} = \prod_{i} (1 - q_{i}^{2})^{ν_{i}}$ in $1 + q Z [⁣ [q] ⁣]$ . Hence $B_{L} (x, x) \in 1 + q Z [⁣ [q] ⁣]$ if and only if $B_{K} (x, x) \in 1 + q Z [⁣ [q] ⁣]$ ; since a rational function with a power-series expansion has no pole at zero, the signed norm condition $B_{L} (x, x) \in 1 + q Z [⁣ [q] ⁣]$ implies $B_{K} (x, x) \in A_{0}$ .

1.2F1F2

The signed set contains $\pm G_{\infty}$ . Each $G_{\infty} (b)$ is integral and bar fixed by [F2]; its $B_{K}$ -norm belongs to $A_{Z}$ and is $1$ modulo $q$ , so $B_{L} (G_{\infty} (b), G_{\infty} (b)) = C_{ν} B_{K} (G_{\infty} (b), G_{\infty} (b)) \in 1 + q Z [⁣ [q] ⁣]$ . Thus both signs of every lower global vector satisfy the source signed norm condition.

2.1F1F2step 1.1step 1.2algebra

Conversely, the signed set is contained in $\pm G_{\infty}$ . Let $x \in U_{A}^{-}$ be bar fixed with $B_{L} (x, x) \in 1 + q Z [⁣ [q] ⁣]$ , and write its finite homogeneous decomposition $x = \sum_{ν} x_{ν}$ . For each nonzero $x_{ν}$ choose its zero-chart order $t_{ν} \in Z$ , so $y_{ν} = q^{- t_{ν}} x_{ν} \in L (\infty)_{- ν} ∖ q L (\infty)_{- ν}$ . By [F2], $B_{K} (y_{ν}, y_{ν})$ is a unit of $A_{0}$ with positive rational residue. Since every $C_{ν}$ has constant term $1$ , the terms $C_{ν} q^{2 t_{ν}} B_{K} (y_{ν}, y_{ν})$ of $B_{L} (x, x)$ with least order have positive leading coefficients and cannot cancel. Regularity therefore forces every $t_{ν} \geq 0$ , so $x \in L (\infty)$ . Bar fixedness puts $x \in ψ (L (\infty))$ , hence $x \in E_{Z} = ⨁_{b} Z G_{\infty} (b)$ , say $x = \sum_{b} a_{b} G_{\infty} (b)$ with integers $a_{b}$ and finite support. Homogeneity gives $B_{L} (x, x) = \sum_{ν} C_{ν} \sum_{b, c \in B (\infty)_{- ν}} a_{b} a_{c} B_{K} (G_{\infty} (b), G_{\infty} (c))$ , whose residue is $\sum_{b} a_{b}^{2}$ by [F2]. This equals $1$ , so exactly one $a_{b}$ is $\pm 1$ and the rest vanish; thus $x = \pm G_{\infty} (b)$ . Together with step 1.2 this proves equality of the full signed sets.

3.1F3step 2.1

String partition and $T_{i, n}$ transition. The source defines $B_{i, n}$ by the left string ideals; write $T_{i, n}$ for its transported map $π_{i; n}$ , which selects the unique signed element of exact string position $n$ whose difference from $F_{i}^{(n)}$ times a lower position is an integral combination of strictly higher string positions. The adapted global bases of [F3] identify that partition with the crystal $ε_{i}$ -partition of $B (\infty)$ , and the exact top-string transition of [F3] gives $T_{i, n} (G_{\infty} (b)) = G_{\infty} ({\tilde{f}}_{i}^{n} b)$ whenever ${\tilde{e}}_{i} b = 0$ , with sign $+ 1$ fixed by the coefficient- $1$ quotient relation.

4.1step 2.1step 3.1

The positive canonical recursion. The source constructs its positive canonical basis from $1$ by the transitions $T_{i, n}$ ; the Kashiwara crystal satisfies the same recursion because every positive-height class $b$ has some $ε_{i} (b) = n > 0$ with ${\tilde{e}}_{i}^{n} b$ of strictly smaller height and $b = {\tilde{f}}_{i}^{n} ({\tilde{e}}_{i}^{n} b)$ . Induction on height using step 3.1 therefore identifies the positive canonical basis with $G_{\infty} (B (\infty))$ , removing the sign ambiguity of step 2.1 by the actual recursion and not merely by the almost-orthonormal norms.

5.1F4F5step 4.1

Highest-module quotient bases. The source defines its highest-module canonical basis by the based quotient of the half, omitting precisely the killed right-string-ideal elements; [F4] gives exactly the same zero-or-survive formula on $G_{\infty}$ , and [F5] matches the highest eigenvalue $v^{⟨ h, λ ⟩} = q^{- ⟨ h, λ ⟩}$ with the action of $K_{- h}$ and discharges the regularity hypotheses. Hence the quotient canonical bases coincide with the lower global bases $G_{λ}$ for every dominant label $λ$ .

6.1step 2.1step 4.1step 5.1∎

Conclusion. Steps 1.1–2.1 prove the signed-set equality ${\pm G_{\infty}}$ ; steps 3.1–4.1 prove the positive basis identification by the actual top-string recursion; step 5.1 proves the highest-module quotient identification under the exact transport dictionary. No upper basis and no unitriangular word characterization was used.

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Sources

Masaki Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Mathematical Journal 63 (1991), 465–516 (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)
Jonathan Brundan, Dual Canonical Bases and Kazhdan–Lusztig Polynomials, arXiv:math/0509700v2 (Transform. Groups 14 (2009), 535–558) (standard reference, not scraped)