Filtrations on block subalgebras of reduced universal enveloping algebras

Let $G$ be a semisimple connected simply connected algebraic group. Assume that the characteristic $p$ of the field is ‘very good’ as in [19]. Throughout the paper, we will need to understand the structure of the center of $\mathcal{U}(\mathfrak{g})$ and its image under the quotient map to $\mathcal{U}_{\chi}(\mathfrak{g})$.

Under our assumptions we may write the center down explicitly using the following theorem:

Furthermore, we may make further characterizations of $Z_{\mathrm{Fr}}^{G}$ and $Z_{\mathrm{HC}}$ - the Harish-Chandra isomorphism allows us to understand $G$-invariant elements through the Cartan subalgebra $\mathfrak{h}$.

When considering the image of $Z(\mathcal{U}(\mathfrak{g}))$ in $\mathcal{U}_{\chi}(\mathfrak{g})$, we may use these results to understand this as well.

Suppose $G$ is a reductive algebraic group. We may decompose $\mathsf{Rep}(G)$ as a direct sum of smaller categories, which creates the block decomposition $\mathcal{U}_{\chi}(\mathfrak{g})=\bigoplus_{\lambda}A_{\lambda}$ that we study. Before stating this decomposition, we need to define the notion of a Serre subcategory.

We note that when we refer to $\mathsf{Rep}(G)_{\lambda}$ we fix a fundamental domain for $\mathfrak{X}/(W^{\mathrm{aff}},\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}_{p})$. This applies to all other block decompositions as well. When explicitly needed, we will give a description of this fundamental domain.

The similar results hold for $\mathsf{Rep}(\mathfrak{g})$.

In the case $\lambda=w\cdot\mu$ for $w\in W$, we say that $\lambda,\mu$ are $W$-linked. Letting $\chi\in\mathfrak{g}^{*}$ be a character of $\mathfrak{g}$, we may pass to the reduced universal enveloping algebra

Through the universal property of $\mathcal{U}(\mathfrak{g})$, we have a bijection

This restricts to $\mathcal{U}_{\chi}(\mathfrak{g})$-modules when we consider $\mathfrak{g}$-modules with $p$-character $\chi$.

Restricting the result of Proposition 2 to the subcategory $\mathsf{Rep}(\mathfrak{g})_{\chi}$ of $\mathfrak{g}$-modules with a given character $p$-character $\chi$, we obtain a block decomposition of $\mathsf{Rep}(\mathfrak{g})_{\chi}$. The simple modules in $\mathsf{Rep}(\mathfrak{g})_{\chi}$ are constructed as the quotients of the so-called baby Verma modules given by $\Delta_{\chi,\lambda}:=\mathcal{U}_{\chi}(\mathfrak{g})\otimes_{\mathcal{U}_{\chi}(\mathfrak{b}^{+})}k_{\lambda}$. It is known that all simple $\mathcal{U}_{\chi}(\mathfrak{g})$-modules arise in this way, see [12].

We have a block decomposition in this case, which is simply an algebraic fact. One may define a linkage relation among principal indecomposable modules for $\mathcal{U}_{\chi}(\mathfrak{g})$, as in [7] §55.

The blocks can again be viewed as Serre subcategories for corresponding simple modules. In the case of $G=\mathrm{SL}_{2}$, we know this decomposition explicitly because there is a complete understanding of the baby Verma modules and simple modules for any character $\chi$.

A more precise result is an explicit description of the blocks in terms of $W$-linkage classes. We may explicitly characterize the blocks by the associated baby Verma modules. We assume $\mathfrak{g}$ is semisimple, and define a modified weight lattice $\mathfrak{X}_{\chi}$ as in [14] §6.2 via

We have $|\mathfrak{X}_{\chi}|=p^{\dim\mathfrak{h}}$. This will index all of the baby Verma modules $\Delta_{\chi,\lambda}$.

We note that the Harish-Chandra center of $\mathfrak{g}$ and, in particular, the elements $c_{1},\ldots,c_{n}$, act by a constant on the simple modules within a given block. For $\mathfrak{sl}_{2}$ this action uniquely determines the block and thus we will sometimes refer to the block by the constant by which the generator of the Harish-Chandra center acts on the simples in the block.

It is worth noting that in characteristic $p$, the differentiation functor

induces an equivalence of categories between $\mathsf{Rep}(G_{1})$ and $\mathcal{U}_{0}(\mathfrak{g})$-modules, so in this case Theorem 2 is an immediate consequence of a block decomposition on $\mathsf{Rep}(G_{1})$. Here, $G_{1}$ is the kernel of the Frobenius twist $\mathrm{Fr}:G\to G^{(1)}$. This illustrates the connection between the two block decompositions.

Given this block decomposition, we may decompose the algebra $\mathcal{U}_{\chi}(\mathfrak{g})$ into blocks associated to each Serre subcategory. In particular, we may write

where nonzero element $x\in A_{\lambda}$ if it acts nontrivially on $\mathsf{Rep}(\mathfrak{g})_{\chi,\lambda}$ and trivially on all the other subcategories in the decomposition of Theorem 2.

For any decomposition of an algebra into direct summands, there exists a corresponding collection of central idempotents. In particular, the decomposition $\mathcal{U}_{\chi}(\mathfrak{g})=\bigoplus_{\lambda}A_{\lambda}$ can be written as

where $\pi_{\lambda}$ are central idempotents. Note that due to the result of [11], the central idempotents of $\mathcal{U}_{\chi}(\mathfrak{g})$ lie in the image of the center of $\mathcal{U}(\mathfrak{g})$. This allows to interpret the projection maps onto each block as multiplication by $\pi_{\lambda}$.

Recall from the introduction that there is a natural $G$-action on $\mathcal{U}_{0}(\mathfrak{g})$ induced by the adjoint action of $G$. Since the central idempotents $\pi_{\lambda}$ are $G$-invariant, the corresponding projections are $G$-equivariant and the adjoint action preserves the subalgebras $A_{\lambda}$.

In this subsection, we briefly review the representation theory of $\mathrm{SL}_{2}$ and $\mathfrak{sl}_{2}$ in characteristic $p$. A good reference for this is [18]. This is also summarized (on a less detailed level) in [12] and [14]. Throughout this subsection, we will need $p>2$ for these results to hold.

The flag variety of $\mathrm{SL}_{2}$ is $\mathbf{P}^{1}$. This group has weight lattice $\mathfrak{X}\simeq\mathbf{Z}$ and $\rho=1$. We construct representations of $\mathrm{SL}_{2}$ known as Weyl modules - these are defined as $\nabla_{\lambda}=\mathrm{H}^{0}(\mathbf{P}^{1},\mathcal{O}(\lambda))$. We can identify this with

where $x,y$ are homogeneous coordinates for $\mathbf{P}^{1}$ and the subscript denotes the degree $\lambda$ part of the grading on $k[x,y]$. We note that $\nabla_{\lambda}$ are indexed by $\mathfrak{X}_{+}$ and the Weyl modules are defined for a general reductive algebraic group in a similar fashion.

Next, we discuss the representations of $\mathfrak{sl}_{2}$. We first need to choose a basis, which will be fixed for the rest of the paper.

Because we assume $p>2$, the trace form $4\mathrm{tr}(xy)$ is non-degenerate and hence we have $\mathfrak{sl}_{2}\simeq\mathfrak{sl}_{2}^{*}$. Thus, when classifying characters we may consider the associated elements in $\mathfrak{sl}_{2}$. We may divide them into three categories, up to conjugacy by $\mathrm{SL}_{2}$:

For each of these cases, we have a corresponding classification of simple $\mathcal{U}_{\chi}(\mathfrak{sl}_{2})$ modules, or simple $\mathfrak{sl}_{2}$-modules with $p$-character $\chi$. To distinguish from the notation for $\mathrm{SL}_{2}$-modules, we add the subscript $\chi$ to denote which character we are using. We state with these the grouping into different blocks following Theorem 2.

In our computations, we will also need some understanding of the projective covers of $L_{\chi,\lambda}$.

The representations of $\mathrm{SL}_{2}$ and $\mathfrak{sl}_{2}$ are related by differentiation functor in the following way.

Using the following lemma, we can determine bases for $\mathcal{U}_{\chi}(\mathfrak{sl}_{2})$ and write down the kernel of $\mathcal{U}(\mathfrak{g})\to\mathcal{U}_{\chi}(\mathfrak{g})$.

Finally, we discuss the center of $\mathcal{U}(\mathfrak{sl}_{2})$. As discussed above, this is generated by $Z_{\mathrm{Fr}}$ and $Z_{\mathrm{HC}}$. The lemma above essentially tells us the form of $Z_{\mathrm{Fr}}$. As for $Z_{\mathrm{HC}}$, we observe that for $\mathfrak{sl}_{2}$ the Cartan subalgebra $\mathfrak{h}$ is simply the span of $h$. Then we have $\mathrm{S}(\mathfrak{h})\simeq k[h]$, and thus

The finite Weyl group for $\mathfrak{sl}_{2}$ is isomorphic to $\mathbf{Z}/2\mathbf{Z}$, and the dot action sends $h\mapsto-h-2$. Thus, $Z_{\mathrm{HC}}$ is isomorphic to $k[(h-1)^{2}]$, and has a single generator. For our convenience we will denote $h-1$ by $h^{\prime}$.

We will be discussing three filtrations of interest in this section. First, recall the definition of the PBW filtration.

If we choose a basis $x_{i}$ of $\mathfrak{g}$ then we have monomial basis $\{\prod_{i}x_{i}^{e_{i}}\}$ of $\mathcal{U}(\mathfrak{g})$ as a vector space over $k$ by the PBW theorem. Then $F_{d}$ is the subspace of $\mathcal{U}(\mathfrak{g})$ spanned by elements $\prod_{i}x_{i}^{e_{i}}$ with $\sum_{i}e_{i}\leq d$. The filtration is compatible with multiplication in a sense that $F_{i}\cdot F_{j}\subset F_{i+j}$ and is uniquely characterized by this property together with $F_{0}=k\cdot 1,F_{1}=k\cdot 1+\mathfrak{g}\subset\mathcal{U}(\mathfrak{g})$. It is also preserved by the adjoint action of $G$ on $\mathcal{U}(\mathfrak{g})$. Finally, for the associated graded algebra of this filtration we have an isomorphism $\operatorname{{gr}}\mathcal{U}(\mathfrak{g})\simeq S(\mathfrak{g})$.

We obtain a filtration on $\mathcal{U}_{\chi}(\mathfrak{g})$ by applying the quotient map to each vector space in the filtration. From this point onward, we will denote this PBW filtration on $\mathcal{U}_{\chi}(\mathfrak{g})$ by $V_{d}$. The associated graded algebra of this filtration is isomorphic to $k[\mathfrak{g}_{p}]$, where $\mathfrak{g}_{p}$ is the Frobenius neighbourhood of $0$ inside $\mathfrak{g}$, which is defined to be the preimage of $0$ under the Frobenius map $\mathrm{Fr}:\mathfrak{g}\to\mathfrak{g}^{(1)}$. Inside $\mathfrak{g}$ it is given by the ideal generated by the elements $x_{i}^{p}$, where $x_{i}$ runs through a basis of $\mathfrak{g}$.

We note that both filtrations are compatible with multiplication and the pushforward filtration is uniquely characterized by this together with $V_{0}^{\mathrm{pf}}=k\cdot\pi_{\lambda}$ and $V_{1}^{\mathrm{pf}}=k\cdot\pi_{\lambda}+\pi_{\lambda}\mathfrak{g}$.

The last filtration we consider is a shifted PBW filtration.

This last filtration is also compatible with multiplication and is uniquely characterized by this together with $V_{0}^{\mathrm{sh}}=k\cdot\pi_{\lambda},V_{1}^{\mathrm{sh}}=k\cdot\pi_{\lambda}+\pi_{\lambda}\mathfrak{g}+\mathrm{span}_{j}(\pi_{\lambda}c_{j})$.

The associated graded algebras for these on $A_{\lambda}$ are denoted by $\operatorname{{gr}}_{\mathrm{pf}}A_{\lambda},\operatorname{{gr}}_{\mathrm{int}}A_{\lambda},\operatorname{{gr}}_{\mathrm{sh}}A_{\lambda}$ respectively. Finally, note that for $\chi=0$ all three filtrations as well as $V_{i}$ are preserved by the $G$-actions.

In this subsection we study the relation between the nilpotent cone $\mathcal{N}$ and the associated graded algebra of $A_{\lambda}$ with respect to the shifted PBW filtration $V_{i}^{\mathrm{sh}}$.

Throughout this subsection, we let $G$ be a semisimple connected simply connected linear algebraic group with Lie algebra $\mathfrak{g}$. We additionally impose the following assumptions on $G$:

1. (1)

   The derived group $\mathcal{D}G$ of $G$ is simply connected.

2. (2)

   $p$ is odd and a good prime for $G$ (as in §2.1)

3. (3)

   The Killing form on $\mathfrak{g}$ is non-degenerate.

Recall that $\mathcal{D}G$ is the intersection of all normal algebraic subgroups $N\leq G$ so that $G/N$ is commutative. It can be viewed as the algebraic group analogue to the commutator subgroup $[G,G]$. With these conditions on $G$, we may use the relevant results for $G$ - we note that for $G=\mathrm{SL}_{2}$, all these conditions apply, so any theorems of this section may be used for $\mathfrak{sl}_{2}$ in characteristic $p>2$.

Under our assumptions on $G$ the set of closed points of $\mathcal{N}$ is the set of nilpotent elements of $\mathfrak{g}$ (see for example Theorem 2.6, [9]). Note also, that the adjoint $G$-action preserves $\mathcal{N}$.

As we work in characteristic $p>0$, we have a morphism

We define the Frobenius neighborhood of $0$ in $\mathcal{N}$, denoted by $\mathcal{N}_{p}$, to be the preimage of a point $0\in\mathcal{N}^{(1)}$ under this map. This can be written as $\mathcal{N}_{p}\simeq\operatorname{{Spec}}k[\mathfrak{g}_{p}]/\langle c_{1},\ldots,c_{n}\rangle$.

For this subsection, we fix a block $A_{\lambda}$ throughout. Let $\alpha_{i}$ be the constant by which the generator of the center $c_{i}$ acts on the simple modules associated to the block $A_{\lambda}$.

We put $C_{\lambda}$ for the image of the center of $\mathcal{U}(\mathfrak{g})$ in $A_{\lambda}$, which is known as the algebra of coinvariants.

For a root $\mu$ of $G$ let $\mathfrak{g}_{\mu}$ be a corresponding root space. For a positive root $\mu$ of $G$ we let $h_{\mu}\in[\mathfrak{g}_{\mu},\mathfrak{g}_{-\mu}]$ to be the unique element of $\mathfrak{h}$ so that $\mu(h_{\mu})=2$ (recall $p>2$). We now have the following lemma, which is a combination of results from [19] and [3].

In either case, the above Lemma implies that the inclusion $C_{\lambda}\to A_{\lambda}$ after passing to associated graded algebra becomes an injective map $C_{\lambda}\to\operatorname{{gr}}_{\mathrm{sh}}A_{\lambda}$ of graded algebras. In the situation of case (2) we have $V_{i}^{\mathrm{sh}}=V_{i}^{\mathrm{pf}}$, since the coinvariant algebra is trivial and so $\pi_{\lambda}c_{j}$ is a scalar element of $A_{\lambda}$.

Since we have $\pi_{\lambda}(V_{i})\subset V_{i}^{\mathrm{sh}}$, there is a well defined map $\operatorname{{gr}}\mathcal{U}_{\chi}(\mathfrak{g})\to\operatorname{{gr}}_{\mathrm{sh}}A_{\lambda}$. Note that, since $G$ is semisimple, the images of the central elements $c_{i}$ have smaller degree with respect to the shifted filtration then with respect to pushforward filtration and, thus, the image of the ideal $\langle c_{1},\ldots,c_{n}\rangle$ in $\operatorname{{gr}}\mathcal{U}_{\chi}(\mathfrak{g})$ lies in the kernel of the map. It follows that the above map quotients through $k[\mathcal{N}_{p}]\simeq k[\mathfrak{g}_{p}]/\langle c_{1},\ldots,c_{n}\rangle$ since $k[\mathfrak{g}_{p}]\simeq\operatorname{{gr}}\mathcal{U}_{\chi}(\mathfrak{g})$. We then obtain a map

Furthermore, recall from the end of §2.2 that if $\chi=0$ there are $G$-actions on $\mathcal{U}_{\chi}(\mathfrak{g})$ and $A_{\lambda}$ and the map $\mathcal{U}_{\chi}(\mathfrak{g})\to A_{\lambda}$ is $G$-equivariant and then so is the map $k[\mathcal{N}_{p}]\to\operatorname{{gr}}_{\mathrm{sh}}A_{\lambda}$.

Putting these maps together, we obtain a map from $k[\mathcal{N}_{p}]\otimes_{k}C_{\lambda}$ to $\operatorname{{gr}}_{\mathrm{sh}}A_{\lambda}$. We may then show the following theorem.

Given that $C_{\lambda}$ is by definition a quotient of a polynomial $k$-algebra $Z_{\mathrm{HC}}\simeq\mathrm{S}(\mathfrak{h})^{W,\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}}$, we may interpret its spectrum as a subscheme $X_{\lambda}$ of the affine space $\mathfrak{h}/W$. Given that $\mathcal{N}$ and hence $\mathcal{N}_{p}$ are affine schemes, we can interpret the tensor product as a fibre product of these schemes. The surjection onto the associated graded algebra tells us that

can be realized as a closed subscheme of $\mathcal{N}_{p}\times_{k}X_{\lambda}$. Note that for case (2) of Lemma 8 $X_{\lambda}$ is a point and for a nilpotent character it is the spectrum of a local ring, so it has a unique closed point.

Let $W^{\lambda}$ be the set of minimal coset representatives for the subgroup $W(\lambda)$ (the stabilizer of $\lambda$). By [3] §3.19, we, in addition, know the Hilbert series for $C_{\lambda}$, namely

In particular, we have $\dim C_{\lambda}=[W(\lambda+\mathfrak{X}):W(\lambda)]$.

In the case of $\mathfrak{g}=\mathfrak{sl}_{2}$ the scheme $X_{\lambda}$ is equal to $\operatorname{{Spec}}k$ or $\operatorname{{Spec}}k[\varepsilon]/\varepsilon^{2}$, with $\mathcal{N}_{p}\times_{k}X_{\lambda}$ being either $\mathcal{N}_{p}$ or a first order thickening of it. Geometrically, we aim to explicitly describe the ideal sheaves of $\operatorname{{gr}}A_{\lambda}$ as subschemes in the case of $\chi=0$. This is the same as determining the kernel of the map in Theorem 3, which in the $\chi=0$ case is $G$-invariant. Once we decompose the ring of functions on the nilpotent cone into $G$-modules we may compute $k[\mathcal{N}_{p}]\otimes_{k}C_{\lambda}$ as a $G$-module.

In this section we show that the filtrations $V_{i}^{\mathrm{pf}}$ and $V_{i}^{\mathrm{int}}$ have a simple relationship, allowing us to only focus on the pushforward filtration.

The precise relation between these filtrations requires us to provide a non-degenerate associative bilinear form

In [2], the case $\chi=0$ is treated. In [10] it is generalized to arbitrary $\chi$. We will sketch the construction of this bilinear form in the following lemma.

Let $x_{i}$ be a basis for $\mathfrak{g}$, so $\prod_{i}x_{i}^{e_{i}}$ for $e_{i}\in\{0,1,\ldots,p-1\}$ form a basis for $\mathcal{U}_{0}(\mathfrak{g})$. We define $S$ as the span of $\prod_{i}x_{i}^{e_{i}}$ where the $e_{i}$ are not all $p-1$. The bilinear form $b$ constructed in [2] is defined as $b(u,v)=\varphi_{0}(uv)$, where the linear map $\varphi_{0}$ is given by $\varphi_{0}|_{S}=0$ and $\varphi_{0}(\prod_{i}x_{i}^{p-1})=1$. The same definition works in the general case.

If we let $N=\dim_{k}\mathfrak{g}$ and $V_{i}$ be as in §3.1 we have a filtration of length $N(p-1)+1$ on $\mathcal{U}_{\chi}(\mathfrak{g})$

Here, $0$ denotes the trivial vector space. From the competibility of the filtration with multiplication and the definition of the bilinear form $b$ we see that

where $\perp$ denotes the orthogonal complement with respect to $b$.

Additionally, suppose we have a block decomposition $\mathcal{U}_{\chi}(\mathfrak{g})=\bigoplus_{\lambda}A_{\lambda}$. The $A_{\lambda}$ are mutually orthogonal with respect to the algebra structure of $\mathcal{U}_{\chi}(\mathfrak{g})$, and hence we deduce that the bilinear form $b$ restricts to $b_{\lambda}:A_{\lambda}\times A_{\lambda}\to k$ which is still a non-degenerate associative pairing. We denote the orthogonal complement in $A_{\lambda}$ with respect to $b_{\lambda}$ by $\perp_{\lambda}$.