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The type-A nilpotent pyramid, good grading and Whittaker character

Definition

Work over $C$ . Let $λ = (0 \leq p_{1} \leq \dots \leq p_{n})$ be a partition of $N \geq 0$ with transpose $q_{1} \geq \dots \geq q_{l}$ , where $l = p_{n}$ for a nonempty row list and $l = 0$ for an empty row list. Zero-length rows have no boxes. Let $g = {g l}_{N} (C)$ with matrix units $e_{i, j}$ in the standard basis $v_{1}, \dots, v_{N}$ (Classical complex matrix Lie algebras). Number the boxes of the diagram of $λ$ by $1, \dots, N$ down columns from the left, and for a box $i$ write $r o w (i), c o l (i)$ for its row and column, $L (i)$ for the box immediately to its left (or $\emptyset$ if absent), and $R (i)$ for the box immediately to its right (or $\emptyset$ if absent). Define the nilpotent endomorphism $e = \sum_{i : L (i) \neq \emptyset} e_{L (i), i} \in g, e v_{i} = v_{L (i)},$ and declare $\deg e_{i, j} = c o l (j) - c o l (i)$ for all $i, j$ , so that $g = ⨁_{r \in Z} g_{r}$ with $g_{r} = span {e_{i, j} : c o l (j) - c o l (i) = r}$ . Put $p = ⨁_{r \geq 0} g_{r}, h = g_{0}, m = ⨁_{r < 0} g_{r},$ and for rows $1 \leq i, j \leq n$ define the shift $s_{i, j} = p_{j} - p_{i}$ if $i \leq j$ and $s_{i, j} = 0$ if $i \geq j$ , together with the centralizer elements $e_{i, j; r} = \sum_{\begin{array}{c} r o w (h) = i, r o w (k) = j \\ c o l (k) - c o l (h) = r \end{array}} e_{h, k} (s_{i, j} \leq r < p_{j}) .$ Finally let $χ ⁣ : m \to C$ be $χ (x) = tr (x e)$ , where the trace is the trace of the $N \times N$ matrix.

The verification below records the properties of this data that are used throughout this page. $e \in g_{1}$ has Jordan type $λ$ ; $p$ is a parabolic subalgebra with Levi factor $h ≅ {g l}_{q_{1}} \oplus \dots \oplus {g l}_{q_{l}}$ and nilradical $⨁_{r > 0} g_{r}$ , while $m$ is the nilradical of the opposite parabolic; the elements $e_{i, j; r}$ with $s_{i, j} \leq r < p_{j}$ form a basis of the centralizer $g^{e} = {X \in g : [X, e] = 0}$ ; the box set $J = {(a, b) \in {1, \dots, N}^{2} : c o l (a) \leq c o l (b), R (b) = \emptyset}$ is in bijection with $K = {(i, j, r) : 1 \leq i, j \leq n, s_{i, j} \leq r < p_{j}}$ via $(a, b) \mapsto (r o w (a), r o w (b), c o l (b) - c o l (a))$ , and $ξ_{a, b} : = \sum_{t = 0}^{c o l (a) - 1} e_{L^{t} (a), L^{t} (b)} = e_{r o w (a), r o w (b); c o l (b) - c o l (a)};$ and $χ$ is a Lie algebra character, $χ ([m, m]) = 0$ . This grading is the type-A pyramid good grading in the degree-one convention: $ad e : g_{r} \to g_{r + 1}$ is injective for $r < 0$ and surjective for $r \geq 0$ . For $e \neq 0$ it differs from the standard Dynkin grading, in which $e$ has degree $2$ rather than $1$ . For $e = 0$ all boxes lie in one column or there are no boxes, and the pyramid grading and the trivial Dynkin grading both concentrate $g$ in degree zero. If $N = 0$ , take ${g l}_{0} = 0$ : the box set and both centralizer index sets are empty, $e = 0$ , $p = h = m = 0$ , and $χ = 0$ . These conventions also permit an empty row list.

Facts & Assumptions

Given: A partition $λ = (0 \leq p_{1} \leq \dots \leq p_{n})$ of $N$ with transpose $q_{1} \geq \dots \geq q_{l}$ , the diagram boxes with row, column, $L$ and $R$ data, the nilpotent $e = \sum e_{L (i), i}$ , the grading $\deg e_{i, j} = c o l (j) - c o l (i)$ and the subspaces $p, h, m$ , and the elements $e_{i, j; r}$ and $χ$ .

[F1]

Matrix units satisfy $e_{h, k} e_{k, l} = e_{h, l}$ and $e_{h, k} e_{k^{'}, l} = 0$ for $k \neq k^{'}$ , and the trace is $tr (e_{h, k}) = δ_{h k}$ (Classical complex matrix Lie algebras).

[F2]

Jordan strings: a nilpotent endomorphism is described by its strings $w, N w, N^{2} w, \dots$ ; the Jordan type is the partition of the string lengths, and the centralizer of a nilpotent of type $μ$ has dimension $\sum_{h, l} \min (μ_{h}, μ_{l})$ . Indeed its string-pair block is ${Hom}_{C [t]} (C [t] / t^{b}, C [t] / t^{a})$ ; the image of $1$ is any vector killed by $t^{b}$ , a subspace of dimension $\min (a, b)$ (Every finite-dimensional nilpotent endomorphism has a basis of Jordan strings, Nilpotent endomorphisms and their nilpotency index).

[F3]

The trace form $tr (X Y)$ on ${g l}_{N} (C)$ is symmetric and invariant, and the degree grading is multiplicative: if $x \in g_{r}$ and $y \in g_{s}$ then $x y \in g_{r + s}$ , and $tr (x) = 0$ for $x \in g_{r}$ with $r \neq 0$ (Trace forms are symmetric and invariant, Classical complex matrix Lie algebras).

[F4]

The enveloping algebra and representations of $g$ (Universal enveloping algebra, Representations of Lie algebras).

Proof

technique · direct

Given: The pyramid data of the Definition.

1.1F1F2F4algebra

The endomorphism $e$ acts on the box basis by $e v_{i} = v_{L (i)}$ and $e v_{i} = 0$ for boxes with $L (i) = \emptyset$ ; hence the $v_{i}$ in one row form a Jordan string of length $p_{i}$ , because left shifts run from the rightmost box of the row to the first. Consequently $e$ is nilpotent of Jordan type $λ = (p_{1}, \dots, p_{n})$ , $e \in g_{1}$ , and the number of columns is $l = p_{n}$ when the row list is nonempty, with $l = 0$ for the empty row list. Zero-length rows contribute no Jordan strings; for $N = 0$ all box-indexed sums and vector spaces are empty or zero.

1.2F1algebra

Parabolic structure. The degree defines a $Z$ -grading of the associative algebra: $\deg (e_{h, k} e_{k^{'}, l}) = \deg e_{h, k} + \deg e_{k^{'}, l}$ when $k = k^{'}$ by [F1], and $[g_{r}, g_{s}] \subseteq g_{r + s}$ . Hence $p = ⨁_{r \geq 0} g_{r}$ is a subalgebra, its degree-zero part $h$ consists of the matrices $e_{h, k}$ with $c o l (h) = c o l (k)$ , which is exactly the block-diagonal subalgebra with blocks of sizes $q_{1}, \dots, q_{l}$ , so $h ≅ {g l}_{q_{1}} \oplus \dots \oplus {g l}_{q_{l}}$ ; the strictly positive part $⨁_{r > 0} g_{r}$ is a nilpotent ideal and $m = ⨁_{r < 0} g_{r}$ is the nilradical of the opposite parabolic, with $g = m \oplus h \oplus ⨁_{r > 0} g_{r}$ .

1.3F2algebra

The centralizer basis. First $[e, e_{h, k}] = e_{L (h), k} - e_{h, R (k)}$ for boxes $h, k$ , with a term omitted when the relevant neighbour is absent (this is $[e, e_{h, k}] = \sum_{d} (e_{L (d), d} e_{h, k} - e_{h, k} e_{L (d), d})$ and $e_{L (d), d} e_{h, k} = δ_{d, h} e_{L (d), k}$ , while $e_{h, k} e_{L (d), d} = δ_{L (d), k} e_{h, d}$ and $L (d) = k$ holds exactly for $d = R (k)$ in the same row). Now write $e_{i, j; r} = \sum_{c} e_{h_{c}, k_{c}}$ with $h_{c}$ the box of row $i$ in column $c$ and $k_{c}$ the box of row $j$ in column $c + r$ , the sum over $c$ with $1 \leq c \leq p_{i}$ and $1 \leq c + r \leq p_{j}$ ; here $r \geq s_{i, j} \geq 0$ by the defining range, so this index set is exactly $c = 1, \dots, \min (p_{i}, p_{j} - r)$ . Then $[e, e_{i, j; r}] = \sum_{c} (e_{h_{c - 1}, k_{c}} - e_{h_{c}, k_{c + 1}})$ where a term is omitted when the neighbour does not exist; shifting the second sum by $c \mapsto c + 1$ and using $R (k_{c}) = k_{c + 1}$ (which exists exactly when $c + r < p_{j}$ ) shows that the two sums run over the same index set ${c : c \geq 2, c \leq \min (p_{i}, p_{j} - r)}$ , because for every $c$ in the defining range one has $c + r \leq p_{j}$ ; hence the commutator is $0$ . The $e_{i, j; r}$ are linearly independent because they are sums over pairwise disjoint matrix units, and their number is $\sum_{i, j} # {r : s_{i, j} \leq r < p_{j}} = \sum_{i, j} (p_{j} - \max (0, p_{j} - p_{i})) = \sum_{i, j} \min (p_{i}, p_{j})$ , which equals $\dim g^{e}$ by [F2]. Hence they form a basis of $g^{e}$ .

2.1step 1.3algebra

The bijection $J \to K$ . For $(a, b) \in J$ put $i = r o w (a)$ , $j = r o w (b)$ and $r = c o l (b) - c o l (a)$ . Because $b$ is rightmost, $c o l (b) = p_{j}$ ; hence $r = p_{j} - c o l (a) \geq 0$ , $r < p_{j}$ because $c o l (a) \geq 1$ , and $r \geq p_{j} - p_{i}$ because $c o l (a) \leq p_{i}$ . Thus $s_{i, j} = \max (0, p_{j} - p_{i}) \leq r < p_{j}$ , so the map lands in $K$ . The triple determines $b$ as the rightmost box of row $j$ and $a$ as the box of row $i$ in column $p_{j} - r$ , proving injectivity. Conversely, if $(i, j, r) \in K$ , then $r < p_{j}$ gives $1 \leq p_{j} - r$ , while $r \geq \max (0, p_{j} - p_{i})$ gives $p_{j} - r \leq p_{i}$ . Thus those two boxes exist, their columns satisfy $c o l (a) \leq c o l (b)$ , and they give the required preimage. If either row has length zero, the index range in $K$ is empty, consistently with the absence of such a box pair.

2.2F3step 1.2algebra

The character. For $x, y \in m$ one has $[x, y] \in [m, m] \subseteq g_{\leq - 2}$ by step 1.2, and $e \in g_{1}$ , so $[x, y] e \in g_{\leq - 1}$ and $tr ([x, y] e) = 0$ by [F3]. Hence $χ ([m, m]) = 0$ , that is, $χ$ is a Lie algebra character on $m$ : for $x, y \in m$ , $χ ([x, y]) = tr ([x, y] e) = 0$ .

2.3F1F3step 1.1step 1.3algebra

The good-grading conditions. By step 1.3, the centralizer $g^{e}$ has a homogeneous basis $e_{i, j; r}$ with $r \geq s_{i, j} \geq 0$ , so it has no negative-degree part. The kernel of $ad e : g_{r} \to g_{r + 1}$ is $g^{e} \cap g_{r}$ , hence this map is injective for $r < 0$ . The trace pairing is nondegenerate between $g_{r}$ and $g_{- r}$ : each matrix unit $e_{a, b}$ pairs with $e_{b, a}$ , and matrix units of other total degree have zero trace. Invariance gives $tr ([e, x] y) = - tr (x [e, y])$ , so the transpose of the map on degree $r$ is, up to sign, $ad e : g_{- r - 1} \to g_{- r}$ . For $r \geq 0$ the latter is injective by the negative-degree result; finite-dimensional duality therefore makes the original map surjective. These are the good-grading conditions for degree-one $e$ . They remain valid when $e = 0$ , because all nonzero spaces then have degree zero.

3.1step 1.3step 2.1algebra

The identity $ξ_{a, b} = e_{r o w (a), r o w (b); c o l (b) - c o l (a)}$ . Left shifts of $a$ run through the boxes of row $i$ in columns $c o l (a), c o l (a) - 1, \dots, 1$ , and the corresponding left shifts of $b$ run through the boxes of row $j$ in columns $c o l (b), c o l (b) - 1, \dots, c o l (b) - c o l (a) + 1$ ; all these exist because $c o l (b) - c o l (a) + 1 \geq 1$ and the shifts stop while the row $i$ box exists. The pairs $(r o w (h), r o w (k))$ obtained are exactly the pairs $(i, j)$ with column difference $c o l (b) - c o l (a)$ and $c o l (h)$ ranging over $1, \dots, c o l (a)$ ; by step 2.1 these are exactly the pairs indexing $e_{r o w (a), r o w (b); c o l (b) - c o l (a)}$ . Hence the two finite sums coincide, and the two displayed families are the same basis of $g^{e}$ .

4.1F1step 1.1step 1.3step 2.3algebra∎

Dynkin normalization and the zero boundary. If $e \neq 0$ , step 1.1 puts this nonzero element in pyramid degree $1$ , whereas the standard Dynkin grading defined by an ${s l}_{2}$ -triple has $[h, e] = 2 e$ and puts it in degree $2$ ; these two gradings therefore differ. There is no assertion that every Dynkin matrix degree is even: on Jordan strings of lengths one and two the standard $h$ -weights are $0$ and $1, - 1$ , so matrix units between these strings have odd weights. If $e = 0$ , no row has length greater than one, since a row of length at least two contributes a distinct nonzero left-shift matrix unit. Hence all boxes lie in the first column or there are none, every matrix unit has pyramid degree zero, and the zero triple gives the trivial Dynkin grading as well. If $N = 0$ , there are no matrix units or centralizer indices; the parabolic, Levi, opposite nilradical and trace character are zero. Thus the conventions and good-grading conditions cover repeated row lengths, zero-length rows and the empty pyramid without an exclusion.

Remarks

The numbering of boxes down columns from the left makes each row a Jordan string of $e$ ; this is why the row lengths $p_{i}$ (weakly increasing) are the Jordan block sizes, while the column heights $q_{j}$ are the ranks' increments and index the Levi blocks of $h$ .
The shift $s_{i, j} = \max (0, p_{j} - p_{i})$ counts the overlap $\min (p_{i}, p_{j})$ of the two rows $i, j$ through the identity $# {r : s_{i, j} \leq r < p_{j}} = p_{j} - \max (0, p_{j} - p_{i})$ ; this count is used again in the loop-mode index sets of the finite W-algebra items.

Depends on

Used by

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Sources

Jonathan Brundan and Alexander Kleshchev, Schur-Weyl duality for higher levels, Selecta Math. (N.S.) 14 (2008) 1-57 (complete 50-page arXiv article) (standard reference, not scraped)
Jonathan Brundan and Alexander Kleshchev, Representations of shifted Yangians and finite W-algebras, Mem. Amer. Math. Soc. 196 (2008), no. 918 (complete 109-page arXiv memoir) (standard reference, not scraped)