THE $\ell$-MODULAR REPRESENTATION OF REDUCTIVE GROUPS OVER FINITE LOCAL RINGS OF LENGTH TWO

Let $\mathcal{O}$ be a discrete valuation ring with a unique maximal ideal $\mathfrak{p}$ and having finite residue field $\mathbb{F}_{q}$, the field with $q$ elements where $q$ is a power of a prime $p$. We denote $\mathcal{O}_{r}$ to be the reduction of $\mathcal{O}$ modulo $\mathfrak{p}^{r}$. Similarly, given $\mathcal{O}^{\prime}$ a second discrete valuation ring with the same residue field $\mathbb{F}_{q}$, we can define $\mathcal{O}^{\prime}_{r}$. The complex representation theory of general linear groups over the rings $\mathcal{O}_{r}$ has been heavily study [13]. In particular, it has been conjectured by Onn [8] that there is an isomorphism of group algebras $\mathbb{C}[\mathbb{GL}_{n}(\mathcal{O}_{r})]\cong\mathbb{C}[\mathbb{GL}_{n}(\mathcal{O}^{\prime}_{r})]$. When $r=2$, this conjecture was proven by Singla [10]. Moreover, assuming $p$ is odd, Singla proved a generalization of the conjecture for $r=2$ when $\mathbb{G}$ is either $\mathrm{SL}_{n}$ with $p\nmid n$ or the following classical groups $\mathrm{Sp}_{n}$, $\mathrm{O}_{n}$, and $\mathrm{U}_{n}$ [11]. Later on, Stasinski proved it for $\mathrm{SL}_{n}$ for all $p$ [14]. More generally, for a possibly non-classical reductive group, Stasinski and Vera-Gajardo compared the complex representation of the two groups in question [15].

Let us briefly describe the result of Stasinski and Vera-Gajardo. First recall that for any group scheme $\mathbb{G}$ over $\mathbb{Z}$ and any commutative ring $R$, we may speak of the group $\mathbb{G}(R)$ of $R$-points. We now let $\mathbb{G}$ be a reductive group scheme over $\mathbb{Z}$ – for details on such group schemes see e.g. [2] and [3]. Note that for any root datum – see e.g. [3] – there is a split reductive group scheme $\mathbb{G}$ over $\mathbb{Z}$ with this root datum [2, Exp. XXV Theorem 1.1]. The group $\mathbb{G}\times\mathbb{F}_{q}$ obtained by base-change with the finite field $\mathbb{F}_{q}$ of characteristic $p$ is a reductive algebraic group over $\mathbb{F}_{q}$, and we want to insist that

we observe that the condition $\mathbf{(*)}$ depends only on the root datum of $\mathbb{G}$ – see [12, Section 4] for the definition of good/very good primes.

Under assumption $\mathbf{(*)}$, Staskinski and Vera-Gajardo proved that $\mathbb{C}[\mathbb{G}(\mathcal{O}_{2})]\cong\mathbb{C}[\mathbb{G}(\mathcal{O}^{\prime}_{2})]$. When $\mathbb{G}\times\mathbb{F}_{q}$ is a classical absolutely simple algebraic group not of type $A$, any $p>2$ is very good for $\mathbb{G}.$ When $G=\operatorname{SL}_{n}$ or $\operatorname{SU}_{n}$, $p$ is very good for $\mathbb{G}$ if and only if $n\not\equiv 0\pmod{p}.$ If $\mathbb{G}\times\mathbb{F}_{q}$ is absolutely simple of exceptional type, any $p>5$ is very good for $G$; see [12, 4.3] for the precise conditions when $p\leq 5$.

We study the $\ell$-modular representation of such group scheme over $\mathcal{O}_{2}$ a local ring of length two with finite residue field. One can show that $\mathcal{O}_{2}$ is isomorphic to one of the following rings: $\mathbb{F}_{q}[t]/t^{2}$ or $W_{2}(\mathbb{F}_{q})$, the ring of Witt vectors of length two over $\mathbb{F}_{q}$ [15, Lemma 2.1]. Thus, we let $G_{2}=\mathbb{G}(\mathcal{O}_{2})$ and $G^{\prime}_{2}=\mathbb{G}(\mathcal{O}^{\prime}_{2})$, where $\mathbb{G}$ stands for a reductive group scheme over $\mathbb{Z}$ such that $p$ is very good for $\mathbb{G}\times\mathbb{F}_{q}$.

In this paper, we generalize the previous results over $\mathbb{C}$ to results over a sufficiently large field $K$ of characteristic $l\neq p$. More precisely, we prove that there exists an isomorphism of group algebras $KG_{2}\cong KG^{\prime}_{2}$ over a sufficiently large field $K$ of characteristic $l$ as long as $l\neq p$. We take $K$ to be a sufficiently large so that the representation theory of the groups over $K$ is the same as the representation theory over an algebraically closed field of characteristic $l$.

In order for us to understand those two group algebras, we study their decomposition by block algebras. Given $A$ a $K$-algebra, we define a block of $A$ to be a primitive idempotent $b$ in the center of $A$, which is denoted as $Z(A)$; the algebra $Ab$ is called a block algebra of $A$. By an idempotent, we mean an element $b$ such that $b^{2}=b$ and primitive if $b=b_{1}+b_{2}$ is an expression of $b$ as a sum of idempotents such that $b_{1}b_{2}=0$, either $b_{1}=0$ or $b_{2}=0$. Moreover, the block algebra $Ab$ is an indecomposable two-sided ideal summand of $A$. Thus for a finite group $G$, we can write

where the $B_{i}=e_{i}KG$ are the unique block algebras of $KG$ up to ordering [1]. Moreover $e_{i}\cdot e_{j}=0$ whenever $i\neq j$.

We investigate blocks of $KG$ which we denote as $Bl(G)$. To understand the block algebra of those two group algebras, we exploit the fact that those two groups are extensions of $\mathbb{G}(\mathbb{F}_{q})$ by an abelian $p$-group denoted as $N$ which is isomorphic to the Lie algebra of $\mathbb{G}(\mathbb{F}_{q})$ denoted as $\mathfrak{g}$ [15, Lemma 2.3.]. In fact, for the rest of this section let $G$ denote either $G_{2}$ or $G^{\prime}_{2}$. Let the map

be the surjective map obtained from the map $\mathcal{O}_{2}\to\mathbb{F}_{q}$ with $N=ker(\rho)$. The two groups $G_{2}$ and $G^{\prime}_{2}$ act on $\mathfrak{g}$, via its quotient $\mathbb{G}(\mathbb{F}_{q})$. This action is transformed by the above isomoprhism into the action of $G$ on its normal subgroup $N$, we explore the details of this action in section 3. In section 2, we use Clifford’s theory to relate blocks of $G$ with blocks of $N$. Specifically, we have that any block $b\in Bl(G)$ is equal to $Tr^{G}_{H}(d)=\sum\limits_{g\in{G/H}}gdg^{-1}$ for $d$ in $Bl(H)$, where $H$ is the stabilizer in $G$ for some block of $N$. In section 4, we exploit the fact that Clifford theory, in combination with the results obtained in the characteristic zero case, gives an isomorphism between certain blocks algebra of the stabilizer of those two groups. In section 5, we induce the previous isomorphism to an isomorphism between block algebras of $G_{2}$ and $G^{\prime}_{2}$ by taking advantage of the fact that blocks of a group algebra are interior $G$-algebras. This isomorphism will then give rise to an isomorphism between those two group algebras.

Let $G$ be a finite group and $N$ a normal subgroup of $G$. We use Clifford theory to relate blocks of $KG$ with blocks of $KN$. Given $b$ a block of $KG$ and $e$ a block of $KN$, we say $b$ covers $e$ if $be\neq 0$. We denote the set of blocks of $G$ that covers $e$ by $Bl(G|e)$. Note that there is a natural action of $G$ on the set of blocks of $N$ by conjugation. Moreover one can show that for a fixed $b$, the set of blocks $e$ of $KN$ satisfying $be\neq 0$ is a $G$-conjugacy class of blocks of $KN$ [4, Proposition 6.8.2].

The following Clifford theorem for blocks of $G$ holds:

For more information about Clifford theory of blocks consult Linckelmann [4], Nagao [6], or Craven [1]. Thus in order for us to understand the block algebra of those two groups, we start by investigating the block structure of $eKH$ for a fixed block $e\in Bl(N)$, where as above $H$ is the stabilizer of $e$ in $G$, under the conjugation action.

Let $\operatorname{Irr}_{K}(N)$ be the set of irreducible characters of $KN$. Since $l$ does not divide $|N|$ (because N is a p-group), any block of $KN$ is defined by $e=e_{\chi}=\frac{\chi(1)}{|N|}\sum\limits_{g\in N}\chi(g^{-1})g$, for some $\chi$ in $\operatorname{Irr}_{K}(N)$ [7]. Specifically, $KN$ is a semisimple algebra and $eKN$ is a matrix algebra, so it has defect zero. Thus, we can take advantage of the following proposition to show that $eKH\cong eKN\otimes_{K}K^{\alpha^{-1}}L$. For an $\alpha$ in $Z^{2}(G;K^{*})$, we denote $K^{\alpha}G$ as the twisted group algebra of $G$ by $\alpha$. It has basis as $K$-algebra the elements of $G$ and given $x,y\in G$, we define $x\cdot y=\alpha(x,y)xy$ where $xy$ is the product of $x$ and $y$ in $G$. Furthermore, given a different $\beta$ in $Z^{2}(G;K^{*})$ there is a G-graded algebra isomorphism $K^{\alpha}G\cong K^{\beta}G$ if and only if the classes of $\alpha$ and $\beta$ are equal in $H^{2}(G;K^{*})$ [5]. Thus, for the purpose of this paper $\alpha$ is defined up to an element in the second cohomology group $H^{2}(G;K^{*})$ with $G$ acting trivially on $K^{*}$.

In this section, we study the action of $G$ on the kernel $N$ so we can understand the stabilizer $H$ mod $N$ and thus we can take advantage of the Proposition 2.2. The two groups $G_{2}$ and $G^{\prime}_{2}$ act on $\mathfrak{g}$, via its quotient $\mathbb{G}(\mathbb{F}_{q})$. Particularly, there is a natural adjoint action of $\mathbb{G}(\mathbb{F}_{q})$ on $\mathfrak{g}$:

Also there is an automorphism of $\sigma:\mathbb{G}(\mathbb{F}_{q})\to\mathbb{G}(\mathbb{F}_{q})$ given by raising each matrix entry to the $p$-th power. Note that $\sigma$ composed with $\operatorname{Ad}$ gives rise to a second action of $\mathbb{G}(\mathbb{F}_{q})$ on $\mathfrak{g}$. Thus, given $X\in\mathfrak{g}$, we note that the action of $G_{2}=\mathbb{G}(\mathbb{F}_{q}[t]/t^{2})$ and $G^{\prime}_{2}=\mathbb{G}(W_{2}(\mathbb{F}_{q}))$ are as follow:

where $\bar{g}=\rho(g)$, see [15, Section 2.3] for more details. Moreover, we also have actions of both $G_{2}$ and $G^{\prime}_{2}$ on $\mathfrak{g}^{*}=\operatorname{Hom}_{\mathbb{F}_{q}}(\mathfrak{g},\mathbb{F}_{q})$ and $\operatorname{Irr}_{K}(\mathfrak{g})$. Given $F$ an element of $\mathfrak{g}^{*}$ or $\operatorname{Irr}_{K}(\mathfrak{g})$ and $X\in\mathfrak{g}$, we define:

Given a non-trivial irreducible character $\phi:\mathbb{F}_{q}\to K^{*}$ and for each $\beta\in\mathfrak{g}^{*}$, we define the character $\chi_{\beta}\in\operatorname{Irr}(\mathfrak{g})$ by

Note, there is also an isomorphism of $\mathfrak{g}\cong N$ which is $G$-invariant [15, Lemma 2.3.]. Thus, we can identify the characters of $N$ with characters of $\mathfrak{g}$. Given a character $\chi$ of $N$, we define $H_{2}$ and $H^{\prime}_{2}$ to be the stabilizer of $\chi$ in $G_{2}$ and $G^{\prime}_{2}$ respectively. Thus, we have the following lemma about the stabilizer of $e_{\chi}$.

In this section, we prove that there is an isomorphism of blocks of $eKH_{2}$ and $eKH^{\prime}_{2}$, where $e=e_{\chi}$ for $\chi$ in $\operatorname{Irr}_{K}(N)$ such that $H_{2}$ and $H^{\prime}_{2}$ are the stabilizer of $e$ in $G_{2}$ and $G^{\prime}_{2}$ respectively. To prove this isomorphism, we take advantage of Proposition 2.2. In order to do so, we need to understand the cofactors $\alpha$ associated to $eKH_{2}$ and $eKH^{\prime}_{2}$. We prove that the cofactor $\alpha$ is trivial in both cases. For this, we introduce projective ($K$)-representation of $G$ with factor set $\alpha$. By which, we mean a map $X:G\to\mathbb{GL}_{n}(K)$ such that $X(x)X(y)=\alpha(xy)X(xy)$ for all $x,y$ in $G$ , where $\alpha(xy)$ is in $K^{*}$. In fact, one can check that $\alpha$ is in $Z^{2}(G;K^{*})$, where we assume that $G$ acts on $K^{*}$ trivially. Similarly, one can define a projective representation as a $K^{\alpha}G$-module [6]. We let $H$ denote either $H_{2}$ or $H^{\prime}_{2}$. The projective representations of $H$ are closely related to Clifford theory. In fact, we can use proposition 2.2 to show that there is a projective representation $V$ of $H$ that extends $\chi$. By that we mean that $V$ viewed as $KN$-module is isomorphic to the $KN$-module associated to $\chi$.

In order to understand $\alpha$ from proposition 2.2 which is associated to a projective ($K$)-representation of $G$, we study the projective representation of $H$ with factor set $\alpha$ that extends $\chi$ over a field of characteristic zero. To relate them, we need an $l$-modular system. By this we mean a triple $(F;R;K)$ where $F$ is a field of characteristic zero equipped with a discrete valuation, $R$ is the valuation ring in $F$ with maximal ideal $(\pi)$, and $K=R/(\pi)$ is the residue field of $R$, which is required to have characteristic $l$. If both $F$ and $K$ are splitting fields for $G$ we say that the triple is a splitting $l$-modular system for $G$. Note, we need an $l$-modular system so we can relate representation over a field $F$ of characteristic zero to representation over a field $K$ of characteristic $l$. The following lemma shows that given a projective representation over $F$, we can obtain a projective representation over $R$. Notice this lemma is just a generalization of the already known fact over group algebras that can be extended to hold over twisted group algebras. In fact the prove is the same, see [9, Lemma 2.2.2].

Before we introduce the following theorem, recall that a block of defect zero may be defined as a matrix algebra. Moreover, by the following Proposition 4.3 such a block is a ring summand of $KG$ which has a projective simple module.

Most importantly, by Proposition 4.3 a block of defect zero can have only one simple module and there is a unique ordinary simple module that reduces to it. This is used in the proof of the following theorem.

We recall the following result from Lemma 3.3 that the stabilizer $H_{2}$ and $H^{\prime}_{2}$ of $\chi$ are isomorphic mod $N$ i.e. $H_{2}/N\simeq H^{\prime}_{2}/N$. We will denote $L:=H_{2}/N\simeq H^{\prime}_{2}/N$ for this quotient.

The following results about interior $G$-algebras will be useful in order to prove the isomorphism of those two group algebras. Note first that a $G$-algebra over a field $K$ is an $K$-algebra $A$ together with an action of $G$ on $A$ by $K$-algebra automorphisms. An interior $G$-algebra is a $G$-algebra where the action of $G$ is given by inner automorphism. The example to keep in mind is that $KG$ and $bKG$ are interior $G$-algebra with $G$ acting by conjugation, where $b\in Bl(G)$.

Given $H$ a subgroup of $G$ and $B$ an interior $H$-algebra, we define $\mathrm{Ind^{G}_{H}}(B)$ to be the $K$-module $KG\otimes_{KH}B\otimes_{KH}KG$ and one can put an interior $G$-algebra structure on $\mathrm{Ind^{G}_{H}}(B)$. For more details on the interior $G$-algebra structure on $\mathrm{Ind^{G}_{H}}(B)$ one can consult Thévenaz’s book on $G$-algebras and modular representation theory [16]. In fact, the following lemma shows that as a $K$-algebra one can think of $\mathrm{Ind^{G}_{H}}(B)$ as a matrix algebra over $B$.

In the following proposition, we will see that given certain conditions there is a way of relating the algebra obtained by the induction from $H$ to $G$ of a certain $H$-algebra with the algebra obtained by the idempotent $Tr^{G}_{H}(i)$ where $i$ is an idempotent fixed by $H$. Given $A$ an interior $G$-algebra, we will denote the set of elements of $A$ fixed by $H$ as $A^{H}$ and $1_{A}$ as the multiplicative identity of $A$.

We use Proposition 5.2 to prove the following:

With the above $K$-algebra isomorphism, we define a map $\Psi:KG_{2}\to KG^{\prime}_{2}$ such that $\Psi(a)=\sum\limits_{b\in Bl(G_{2})}\hat{\Phi}(a\cdot b)$ for $a\in KG_{2}$ and show that this map is an isomoprhism of $K$-algebras.