Minimal idempotents on solvable groups

Abstract

In this paper, we begin to develop a theory of character sheaves on an affine algebraic group G defined over an algebraically closed field $\mathtt {k}$ of characteristic $p>0$ using the approach developed by Boyarchenko and Drinfeld for unipotent groups. Let l be a prime different from p. Following Boyarchenko and Drinfeld (Sel Math, 2008. doi:10.1007/s00029-013-0133-7, arXiv:0810.0794v1), we define the notion of an admissible pair on G and the corresponding idempotent in the $\overline{\mathbb {Q}}_l$-linear triangulated braided monoidal category $\mathscr {D}_G(G)$ of conjugation equivariant $\overline{\mathbb {Q}}_l$-complexes (under convolution with compact support) and study their properties. In the spirit of Boyarchenko and Drinfeld (2008), we aim to break up the braided monoidal category $\mathscr {D}_G(G)$ into smaller and more manageable pieces corresponding to these idempotents in $\mathscr {D}_G(G)$. Drinfeld has conjectured that the idempotent in $\mathscr {D}_G(G)$ obtained from an admissible pair is in fact a minimal idempotent and that any minimal idempotent in $\mathscr {D}_G(G)$ can be obtained from some admissible pair on G. In this paper, we prove this conjecture in the case when the neutral connected component $G^\circ \subset G$ is a solvable group. For general groups, we prove that this conjecture is in fact equivalent to an a priori weaker conjecture. Using these results, we reduce the problem of defining character sheaves on general algebraic groups to a special case which we call the “Heisenberg case.” Moreover, as we will see in this paper, the study of character sheaves in the Heisenberg case may be considered, in a certain sense, as a twisted version of the theory of character sheaves on reductive groups as developed by Lusztig (Adv Math 56, 57, 59, 61, 1985, 1986).

Appendix: Torus actions on connected unipotent groups

1.1 Equivariant multiplicative local systems

Let us fix an embedding $\mathbb {Q}_p/\mathbb {Z}_p\hookrightarrow \overline{\mathbb {Q}}_l^*$. Now we can identify isomorphism classes of multiplicative $\overline{\mathbb {Q}}_l$-local systems on any connected (perfect) unipotent group U and central extensions of U by $\mathbb {Q}_p/\mathbb {Z}_p$.

Lemma 10.1

Let U be a connected unipotent group equipped with an action of a torus T such that $U^T=\{1\}$. Then there are no non-trivial T-equivariant multiplicative local systems on U.

Proof

Let $\mathcal {L}$ be a T-equivariant multiplicative local system on U. It corresponds to a central extension⁴ $0\rightarrow A\rightarrow \tilde{U}\rightarrow U\rightarrow 0$ with $\tilde{U}$ also connected and unipotent and A a finite subgroup of $\mathbb {Q}_p/\mathbb {Z}_p$. Since central extensions of a connected group by a discrete group have no non-trivial automorphisms and since $\mathcal {L}$ is T-equivariant, we get an action of T on $\tilde{U}$ making the above central extension T-equivariant. Taking T-fixed points, we get an exact sequence $0\rightarrow A\rightarrow \tilde{U}^T\rightarrow 1.$ Since T is a torus and $\tilde{U}$ is a connected unipotent group, $\tilde{U}^T$ is connected (by [13, §18]). Hence we must have $A\cong 0$, and hence $\mathcal {L}\cong \overline{\mathbb {Q}}_l.$ $\square$

If a torus T acts on a connected unipotent group U, then it follows from [13, §18] that $U^T$ is connected. In this situation we have:

Lemma 10.2

If $\mathcal {L}$ is a T-equivariant multiplicative local system on U such that $\mathcal {L}|_{U^T}\cong \overline{\mathbb {Q}}_l$, then $\mathcal {L}\cong \overline{\mathbb {Q}}_l$. In other words, the restriction homomorphism $(U^*)^T\rightarrow (U^T)^*$ of perfect commutative unipotent groups is an injection.

Proof

Let $\mathcal {L}$ be any T-equivariant multiplicative local system. As before, $\mathcal {L}$ comes from a T-equivariant central extension $0\rightarrow A\rightarrow \widetilde{U}\rightarrow U\rightarrow 0$ with $\widetilde{U}$ connected. Let us consider the central extension obtained by pullback $0\rightarrow A\rightarrow \widetilde{U^T}\rightarrow U^T\rightarrow 0$. Let $\widetilde{u}\in \widetilde{U^T}$, that is $\widetilde{u}$ maps to $U^T\subset U$. Hence $\widetilde{u}\cdot t(\widetilde{u}^{-1})\in A$ for every $t\in T$. Now since T is connected and A is a finite group, we must have $\widetilde{u}\in \widetilde{U}^T$. Hence we have $\widetilde{U^T}=\widetilde{U}^T$. Now let $\mathcal {L}$ be such that the restriction of $\mathcal {L}$ to $U^T$ is trivial. Hence the central extension $0\rightarrow A\rightarrow \widetilde{U^T}=\widetilde{U}^T\rightarrow U^T\rightarrow 0$ is trivial. On the other hand, $\widetilde{U}^T$ is connected since $\widetilde{U}$ is connected and T is a torus. Hence we must have $A\cong 0$ or in other words $\mathcal {L}\cong \overline{\mathbb {Q}}_l.$ $\square$

Lemma 10.3

Let a reductive group $\varGamma$ act on a connected unipotent group U. Then we get the action of $\varGamma$ on the variety $U/{U^\varGamma }$. The only fixed point for this action is the trivial coset $U^\varGamma$.

Proof

Suppose $u \in U$ is such that $c_{\gamma }(u)=u\cdot {\gamma }(u^{-1})\in U^\varGamma$ for each ${\gamma }\in \varGamma$. Since the action of $\varGamma$ on $U^\varGamma$ is trivial, we can check that ${\gamma }\mapsto c_{\gamma }(u)$ defines a group homomorphism $\varGamma \rightarrow U^\varGamma$ which must be trivial since $\varGamma$ is reductive and $U^\varGamma$ is unipotent. Hence $u\in U^\varGamma$. $\square$

1.2 Skew-symmetric biextensions equivariant under torus actions

Let us continue to work in the world of perfect algebraic groups. Let $\mathfrak {cpu}$ (resp. $\mathfrak {cpu}^\circ$) denote the category of perfect commutative unipotent groups (resp. perfect connected commutative unipotent groups). We refer to [2, Appendix A.10] for the theory of skew-symmetric isogenies and biextensions of perfect connected commutative unipotent groups and their associated metric groups. Let $V\in \mathfrak {cpu}^\circ$, and let $V\mathop {\rightarrow }\limits ^{\phi }V^*$ be a skew-symmetric isogeny with kernel $K_\phi$. According to [2, 8], there is the metric group $(K_\phi ,\theta )$ associated with the skew-symmetric isogeny $\phi .$ Let a torus T act on V and hence on $V^*$. Suppose that the isogeny $\phi$ is T-equivariant, i.e., we have a T-equivariant exact sequence

$$\begin{aligned} 0\rightarrow K_\phi \rightarrow V\mathop {\rightarrow }\limits ^{\phi }V^*\rightarrow 0. \end{aligned}$$

(56)

Since T is connected, the action of T on $K_\phi$ is trivial and hence $K_\phi \subset V^T$. Again since T is connected and $K_\phi$ is finite, taking T-fixed points of (56) (and using the argument used in proof of Lemma 10.2), we get the exact sequence

$$\begin{aligned} 0\rightarrow K_\phi \rightarrow V^T\rightarrow (V^*)^T\rightarrow 0. \end{aligned}$$

(57)

Note that according to Lemma 10.2, the restriction of multiplicative local systems induces an injection $(V^*)^T\hookrightarrow (V^T)^*.$ Moreover, by the above isogeny, we have the equality $\dim ((V^*)^T)=\dim (V^T)=\dim ((V^T)^*)$. Hence the above inclusion must in fact be an isomorphism $(V^*)^T\mathop {\rightarrow }\limits ^{\cong } (V^T)^*$. In other words, the quotient $(V^T)^*$ of $V^*$ can be identified with the subgroup $(V^*)^T\subset V^*$. Hence the short exact sequence $0\rightarrow (V/V^T)^*\rightarrow V^*\rightarrow (V^T)^*\cong (V^*)^T\rightarrow 0$ splits as $V^*\cong (V^*)^T\oplus (V/V^T)^*.$ Taking duals, we see that the short exact sequence $0\rightarrow V^T\rightarrow V\rightarrow V/V^T\rightarrow 0$ also splits as $V\cong V^T\oplus V/V^T.$ We see from (57) (and the identification $(V^*)^T\cong (V^T)^*$) that the restriction of $\phi$ to $V^T$ is again a skew-symmetric isogeny which has the same metric group $(K_\phi ,\theta )$ associated with it. In fact, we see that we have the following:

Proposition 10.4

Let $V\in \mathfrak {cpu}^\circ$ be equipped with the action of a torus T. Let $V\mathop {\rightarrow }\limits ^{\phi } V^*$ be a T-equivariant skew-symmetric isogeny with associated metric group $(K_\phi ,\theta ).$ Then V splits as $V^T\oplus V'$ where $V'\subset V$ is T-invariant and the isogeny $\phi$ splits into:

• a skew-symmetric isogeny $\phi ^T:V^T\rightarrow (V^T)^*$ with associated metric group $(K_\phi ,\theta )$ and

• a T-equivariant skew-symmetric isomorphism $\phi ':V'\rightarrow (V')^*.$

In this decomposition, T acts trivially on $V^T$ and $V'$ has no nonzero T-fixed points.

1.3 Torus actions on vector groups

Definition 10.5

A group $V\in \mathfrak {cpu}^\circ$ is said to be a vector group if there is an isomorphism $V\xrightarrow {\cong } \mathbb {G}_a^n$ for some nonnegative integer n. We can use such an isomorphism to define an action of $\mathbb {G}_m$ on V using its natural action on $\mathbb {G}_a^n$. This action is defined to be a linear structure on the vector group V. If V, W are vector groups equipped with a linear structure, we say that a group homomorphism $V\rightarrow W$ is linear if it commutes with the chosen $\mathbb {G}_m$-scalings on V, W.

Remark 10.6

Equivalently, $V\in \mathfrak {cpu}^\circ$ is a vector group if and only if $p\cdot V=0$.

The following result is well known (cf. [7, §4])

Theorem 10.7

Let T be a torus acting on a vector group V by group automorphisms. Then there exists a linear structure on V such that T in fact acts by linear automorphisms with respect to this linear structure.

Hence in order to study vector groups equipped with torus actions, it is enough to study linear actions. If V is such a linear representation of T, then we have a decomposition of V into weight spaces $V=\bigoplus {V_\lambda }$. If $V^*$ is the contragradient representation, then we have

$$\begin{aligned} V^*=\bigoplus {(V_\lambda )^*}=\bigoplus {(V^*)_{-{\lambda }}} \end{aligned}$$

gives the decomposition of $V^*$ into weight spaces since we have an identification $(V^*)_{-{\lambda }}=(V_{\lambda })^*$. However, if we have two linear representations V, W of T, we must also study nonlinear T-maps $V\xrightarrow {\phi } W$. It is clear that $\phi (V_0)=\phi (V^T)\subset W_0=W^T$. We still need to see what happens to the other weight spaces.

For a weight $0\ne \lambda$ of V and $\lambda '$ of W, let us consider the induced T-map $V_\lambda \xrightarrow {\phi '} W_{\lambda '}$. For $v\in V_{\lambda }, t\in T$ we have

$$\begin{aligned} \phi '({\lambda }(t)v)=\phi '(tv)=t\phi '(v)={\lambda }'(t)\phi '(v). \end{aligned}$$

(58)

Since ${\lambda }\ne 0$, this implies that the line spanned by v maps to the line spanned by $\phi '(v)$. Hence if $\phi '$ is nonzero, then ${\text {Hom}}_T(\mathbb {G}_{a,{\lambda }}, \mathbb {G}_{a,{\lambda }'})\ne 0,$ where $\mathbb {G}_{a,{\lambda }}, \mathbb {G}_{a,{\lambda }'}$ are the one-dimensional representations of T corresponding to the weights ${\lambda },{\lambda }'$, respectively.

Let $\phi ':\mathbb {G}_{a,{\lambda }}\rightarrow \mathbb {G}_{a,{\lambda }'}$ be a nonzero T-map with ${\lambda }\ne 0$. We can rescale this map so that $\phi '(1)=1.$ Under the inclusion $\mathbb {G}_m\subset \mathbb {G}_a$, we see that $\phi '({\lambda }(t))={\lambda }'(t)$ by putting $v=1$ in (58). Hence we must have $\phi '=\tau ^n$ (where $\tau :\mathbb {G}_a\rightarrow \mathbb {G}_a$ is the Frobenius automorphism) for some (possibly negative) $n\in \mathbb {Z}$ and ${\lambda }'(t)={\lambda }(t)^{p^n}$, i.e., ${\lambda }'=p^n{\lambda }$ in usual additive notation.

Hence we see that ${\text {Hom}}_T(\mathbb {G}_{a,{\lambda }}, \mathbb {G}_{a,p^n{\lambda }})= \{c\tau ^n|c\in \mathtt {k}\}$ if ${\lambda }\ne 0$, ${\text {Hom}}_T(\mathbb {G}_{a,0}, \mathbb {G}_{a,0})={\text {Hom}}(\mathbb {G}_{a}, \mathbb {G}_{a})=k\{\tau ,\tau ^{-1}\}$ and that all other ${\text {Hom}}$’s are zero.

For a weight ${\lambda }:T\rightarrow \mathbb {G}_m$, and a linear representation V of T, let $V_{{\lambda }}^{(p)}=\sum \limits _{n\in \mathbb {Z}}{V_{p^n{\lambda }}}\subset V$. We have $V_0^{(p)}=V_0$. We have proved

Theorem 10.8

Let V be a linear representation of a torus T. Then we have a decomposition

$$\begin{aligned} V=\bigoplus {V_{\lambda }^{(p)}} \end{aligned}$$

into T-stable subspaces. For the contragradient representation the decomposition is given by

$$\begin{aligned} V^*=\bigoplus {(V^*)_{-{\lambda }}^{(p)}}=\bigoplus {\left( V_{{\lambda }}^{(p)}\right) ^*}. \end{aligned}$$

If V, W are linear representations of T and if $\phi :V\rightarrow W$ is a (not necessarily linear) T-map, then $\phi (V_{\lambda }^{(p)})\subset W_{\lambda }^{(p)}$.

1.4 T-stable maximal isotropic subgroups

Let $V\in \mathfrak {cpu}^\circ$ and let $\phi :V\rightarrow V^*$ be a skew-symmetric biextension (not necessarily an isogeny). For a closed connected subgroup $L\subset V$, let $L^\perp \subset V$ be the kernel of the composition $V\xrightarrow {\phi }V^*\rightarrow L^*$ and $L^{\perp \circ }$ its neutral connected component. For example $V^\perp =\ker (\phi ).$ The biextension $\phi$ is an isogeny if and only if $V^{\perp \circ }=0$.

Definition 10.9

We say that a connected subgroup $L\subset V$ is isotropic if $L\subset L^\perp$, or equivalently if $L\subset L^{\perp \circ }$. If L is an isotropic subgroup, then $\phi$ induces a skew-symmetric biextension on the subquotient

$$\begin{aligned} \phi _L: L^{\perp \circ }/L \rightarrow (L^{\perp \circ }/L)^*. \end{aligned}$$

(59)

A maximal isotropic subgroup is an isotropic subgroup that is maximal among all isotropic subgroups. If L is maximal isotropic, then we must have $\dim (L^{\perp \circ }/L)\le 1$. (See [10, Appendix A.2].)

Now suppose that V is equipped with an action of a torus T such that the biextension $\phi :V\rightarrow V^*$ is T-equivariant. Our goal is to prove

Theorem 10.10

In the situation above, there exists a maximal isotropic subgroup $L\subset V$ such that L is T-stable.

We carried out a similar argument in [10, Appendix A.2]. As loc cit, if we have a T-stable isotropic subgroup we can pass to the associated subquotient. Moreover, the assertion is obvious in case $\dim (V)\le 1.$ Hence it only remains to prove:

Proposition 10.11

In the above notation, if $\dim (V)\ge 2$, then there exists a non-trivial T-stable isotropic subgroup.

Proof

By [10, Lem. A.9], we easily reduce to the case $p\cdot V=0$, i.e., V a vector group. Let us choose a linear structure on V such that the action of T is linear. Hence the action of T on $V^*$ is also linear and we can think of $V^*$ as the contragradient linear representation. By Theorem 10.8, we have decompositions

$$\begin{aligned}&\displaystyle V=\bigoplus {V_{\lambda }^{(p)}},\\&\displaystyle V^*=\bigoplus {\left( V_{\lambda }^{(p)}\right) ^*=\bigoplus {(V^*)_{-{\lambda }}^{(p)}}} \end{aligned}$$

and inclusions $\phi \left( V_{\lambda }^{(p)}\right) \subset \left( V_{-{\lambda }}^{(p)}\right) ^*$. Hence if ${\lambda }\ne 0,$ then $V_{-{\lambda }}^{(p)}$ is a (T-stable) isotropic subgroup. Hence if V has nonzero weights, then we get a non-trivial T-stable isotropic subgroup. On the other hand if T acts trivially on V, then any isotropic subgroup is T-stable. Since $\dim (V)\ge 2$ we know that V must have a non-trivial isotropic subgroup (see [2, Appendix A.10] or [10, Prop. A.10]). $\square$