A bijection between Lusztig series induced by inflation

Question

Context: Let $\pi: \widehat{G} \rightarrow G$ be a surjective morphism between connected reductive groups defined over $\mathbb{F}_q$ whose kernel is a central torus. Then $\pi : \widehat{G}^F \rightarrow G^F$ is surjective, where $F$ denotes the Frobenius morphisms inducing the $\mathbb{F}_q$-rational structures. We thus have an inflation map $- \circ \pi: \text{Irr}(G^F) \rightarrow \text{Irr}(\widehat{G}^F)$. We also have a dual morphism $\pi^* : G^* \rightarrow \widehat{G}^*$ which is injective under the above assumptions.

Question: Is the inflation map compatible with Lusztig series in the sense that for a semisimple $s \in (G^*)^F$ and any $\chi$ in the Lusztig series $\mathcal{E}(G^F,[s])$ associated to $s$ we have $\chi \circ \pi \in \mathcal{E}(\widehat{G}^F,[\pi^*(s)])$?

Motivation: It is easy to see that every element of $\mathcal{E}(\widehat{G}^F,[\pi^*(s)])$ is inflated from $G^F$, so if the answer to my question is positve, this would result in a bijection between $\mathcal{E}(G^F,[s])$ and $\mathcal{E}(\widehat{G}^F,[\pi^*(s)])$.

My Approach: It seems natural to me to try to show that inflation is compatible with Lusztig induction, i.e. $(R^G_T \theta) \circ \pi = R^{\widehat{G}}_{\pi^{-1}(T)} (\theta \circ \pi)$ for any rational maximal torus $T$ and any $\theta \in \text{Irr}(T^F)$. To show this one could try to compare the Deligne-Lusztig varieties $Y_U = \{g \in G \:|\: g^{-1}F(g) \in U\}$ and $Y_{\pi^{-1}(U)}$ for $U$ the unipotent radical of a Borel subgroup of $G$ containing $T$. This looks promising to me as $\pi$ induces a surjective morphism $Y_{\pi^{-1}(U)} \rightarrow Y_{U}$ whose fibres are cosets of $\ker(\pi)^F$. As I see it it would suffice to show that this morphism gives an isomorphism between the $\ker(\pi)^F$-quotient of $Y_{\pi^{-1}(U)}$ and $Y_{U}$ but I am unable to prove it.

I tried to mimic the proof of Proposition 13.20 in Representations of Finite Groups of Lie Type by Digne and Michel where the case $s = 1$ is studied for a more general $\pi$. But unfortunately and contrary to what is claimed there the proof given there does not establish an isomorphism like the above as far as I can tell. Any help is appreciated.

Answer 1

Your approach is correct and is proven in the book by Digne-Michel (in fact a more general statement is proven there). Indeed, by Proposition 13.22 in Digne-Michel we know that

$$R_{T\subseteq B}^G(\theta)\circ \pi = R_{\widehat{T}\subseteq \widehat{B}}^{\widehat{G}}(\theta\circ\pi)$$

where $\widehat{T} = \pi^{-1}(T)$ and $\widehat{B} = \pi^{-1}(B)$. Now you just need to show that the conjugacy class of $\theta\circ\pi$ corresponds to the conjugacy class of $\pi^*(s)$ under duality.

By the definition of duality we have an isomorphism $\lambda : X(T) \to Y(T^*)$ between the character group of $T$ and the cocharacter group of $T^* \leqslant G^*$, which is a dual torus. Furthermore, $\lambda$ is compatible with the Frobenius endomorphism.

According to Proposition 13.7 in Digne-Michel there exists a character $\chi \in X(T)$ of $T$ such that $\theta = \chi|_{T^F}$, i.e., $\theta$ is the restriction of $\chi$. Now let $\gamma = \lambda(\chi)$ be the cocharacter corresponding to $\chi$. If we assume that $n \geqslant 1$ is large enough so that $T$ is a split torus with respect to the Frobenius endomorphism $F^n$ then we have $s = N_{F^n/F}(\gamma(\zeta))$ where $\zeta$ is a $(q^n-1)$th root of unity and $N_{F^n/F} : T^* \to T^*$ is the norm map.

Now let us assume that we have chosen an isomorphism $\widehat{\lambda} : X(\widehat{T}) \to Y(\widehat{T}^*)$ extending $\lambda$. In other words, we have $\widehat{\lambda}(\chi\circ\pi) = \pi^*\circ\lambda(\chi)$. Now, clearly $\theta\circ\pi = (\chi\circ\pi)|_{T^F}$ and $\widehat{\lambda}(\chi\circ\pi) = \pi^*\circ\lambda(\chi) = \pi^*\circ\gamma$. As $\pi^*$ clearly commutes with the norm map this gives us that $\theta\circ\pi$ corresponds to $\pi^*(s) = N_{F^n/F}(\pi^*(\gamma(\zeta)))$. Hence

$$R_{T^*}^G(s)\circ \pi = R_{\widehat{T}^*}^{\widehat{G}}(\pi^*(s))$$,

which is what you want.