Drinfeld curves and geometric Mackey Formula

This blog serves as my preparation for my talk at Bloch's seminar workshop on Deligne-Lusztig varieties. I will present a talk about $\ell$-adic cohomology of Drinfeld curves, which are easiest examples of Degline-Lusztig varieties. To begin with, the story dates back to representation theory of finite group of Lie type (by a finite group of Lie type, I mean $G^F$ with $G$ a reductive group over a finite field and $F$ the Frobenious endomorphism). To say why these groups are important, it is the classification of finite simple groups saying that any finite simplegroup belongs to four classes: cyclic, alternating, $26$ sporadic groups, and the others are all of Lie type. 

The central object of this blog is $SL_2(\mathbb{F}_q)$ with $q=p^r$ a prime power. It is a finite group of Lie type whose cardinality is $q(q-1)(q+1)$ (it's a fun exercise to show this in case you do not know). Not only $SL_2(\mathbb{F}_q)$, but the representation theory of any finite group over complex numbers are "well-understood", they are direct sums of irreducible ones and by orthogonality, one has a criterion for irreducibility via the traces of representations. The total number of irreducible representations equals the number of conjugacy classes. Rather than a brute force method, Drinfeld suggested that one can list all representations (the story is indeed more complicated than that but I don't want to dig further) of $SL_2(\mathbb{F}_q)$ can be found in the first $\ell$-adic cohomology of the affine curve $xy^q - x^qy = 1$, which is now called Drinfeld curve. 

Introduction to $\ell$-adic cohomology

This is probably the most notable achievement of the school of Grothendieck; i.e., the development of étale $\ell$-adic cohomology, which leads to the full solution of Weil conjectures. And it found the application in representation theory is what amazes me at first. In what follows, I give a rapid and formal introduction to $\ell$-adic cohomology with some technical emphasizes.

Let $X$ be a scheme, and $R$ be a torsion ring, i.e., $nR = 0$ for some $n > 0$. We have the notion of small etale site of $X$, denoted $X_{et}$, whose objects are etale morphism $U \to X$ and coverings $\left \{f_i \colon U_i \longrightarrow U \right \}$ consist of etale morphisms $f_i$ that are jointly surjective, i.e., $\bigcup f_i(U_i) = U$. The category of sheaves $\mathbf{Sh}(X_{et},R)$ with values in $R$-modules has enough injectives and we can consider derived functor of the global sections functor

$$\mathcal{F} \longmapsto \Gamma(X,\mathcal{F})$$ which is our etale cohomology $H^i_{et}(X,\mathcal{F})$.  Now let $X/k$ be a variety over an algebraically closed field $k$ of characteristic $p > 0$ and $\ell > 0$ be a prime different from $p$. The groups 

$$H^i(X_{et},\mathbb{Q}_{\ell}) \overset{\mathrm{def}}{=} \left(\varprojlim H^i_{et}(X,\mathbb{Z}/\ell^n \mathbb{Z}) \right) \otimes_{\mathbb{Z}_{\ell}} \mathbb{Q}_{\ell}$$ are call $\ell$-adic cohomology groups of $X$. For simplicity, we erase the subscript $et$. Suppose that $X$ is a projective variety over $k$, here are the list properties of $\ell$-adic cohomology.

• (Finiteness) For all $i \geq 0$, $H^i(X,\mathbb{Q}_{\ell})$ are $\mathbb{Q}_{\ell}$-vector spaces of finite dimensions.
• (Cohomological dimension). For $i > 2\dim(X)$, $H^i(X,\mathbb{Q}_{\ell})=0$. 
• (Kunneth formula). For all $i \geq 0$, there exists a natural isomorphism $$H^i(X \times_k Y,\mathbb{Q}_{\ell}) \simeq \bigoplus_{a+b=i} H^a(X,\mathbb{Q}_{\ell}) \otimes_{\mathbb{Q}_{\ell}} H^b(Y,\mathbb{Q}_{\ell})$$ induced by the cup-product. 
• (Poincare duality). There are isomorphisms $H^0(X,\mathbb{Q}_{\ell}) \simeq \mathbb{Q}_{\ell}[\pi_0(X)]$ (where $\pi_0(X)$ denotes the set of connected components of $X$) and $H^{2\dim(X)}(X,\mathbb{Q}_{\ell}) \simeq \mathbb{Q}_{\ell}$. Moreover, for any $0 \leq i \leq 2\dim(X)$, there is a perfect pairing $$H^i(X,\mathbb{Q}_{\ell}) \otimes H^{2\dim(X)-i}(X,\mathbb{Q}_{\ell}) \longrightarrow H^{2\dim(X)}(X,\mathbb{Q}_{\ell}) \simeq \mathbb{Q}_{\ell}.$$  
• (Trace formula). Let $\varphi \colon X \longrightarrow X$ be an endormorphism, then $$(\Gamma_{\varphi} \cdot \Delta) = \sum_i (-1)^i \mathrm{Tr}(\varphi^*,H^i_{et}(X,\mathbb{Q}_{\ell}),$$ where the LHS is the intersection product of the graph $\Gamma_{\varphi}$ and the diagonal $\Delta \colon X \to X \times_k X$. 

There are other deep properties that we do not use here such as lifiting to $\mathbb{C}$ or weak Lefschetz or Riemann hypothesis. One also has a compactly-supported version, denoted $H^i_c(X,\mathbb{Q}_{\ell})$ ($X$ does have to be projective here) and $H^i_c(X) = H^i(X)$ if $X$ proper. More basically, one has

• (Mayer-Vietoris sequence) Let $Z\subset X$ be a closed immersion and $U  = X \setminus Z$ the open complement, then there is a long exact sequence $$\cdots \longrightarrow H^i_c(U,\mathbb{Q}_{\ell}) \longrightarrow H^i_c(X,\mathbb{Q}_{\ell})  \longrightarrow H^i_c(Z,\mathbb{Q}_{\ell}) \longrightarrow H^{i+1}_c(U,\mathbb{Q}_{\ell}) \longrightarrow \cdots$$  
• ($\mathbb{A}^1$-homotopy) $H^i_c(\mathbb{A}^d_k,\mathbb{Q}_{\ell}) \simeq \mathbb{Q}_{\ell}$ for $i = 2d$ and $0$ otherwise. 

Using Mayer-Vietoris and $\mathbb{A}^1$-homotopy, one can show that

Lemma 1. We have

$$H^i_c(\mathbb{G}_m,\mathbb{Q}_{\ell}) =\begin{cases}\mathbb{Q}_{\ell} & \text{if} \ i = 1,2 \\ 0 & \text{otherwise} \end{cases}$$ and $$H^i_c(\mathbb{P}^1, \mathbb{Q}_{\ell}) = \begin{cases} \mathbb{Q}_{\ell} & \text{if} \ i = 0,2 \\ 0 & \text{otherwise}  \end{cases}.$$ (we note that there should be some Tate twists, which is ignored here). 

Proof. Exercise.

When $k$ is not algebraically, for instance, when $k = \mathbb{F}_q$, then we base change to $\overline{k}$. We make a convention that variety over $\mathbb{F}_q$ is denoted by $X_0$ while their base change to algebraic closures is denoted by $X$. Now if $q = p^r$ and $\ell \neq p$, the variety $X_0/\mathbb{F}_q$ is endowed with a geometric Frobenius morphism $F \colon X \longrightarrow X$ which is identity on the underlying topological space and is $p$-power on structural sheaf. For instance, if $X = \mathbb{A}^1_k$ then $F$ is the dual of $k[t] \longrightarrow k[t], t \longmapsto t^p$ (fix coefficients). The trace formula can be rewritten as

$$\left |X^F  \right| = \sum_{i \geq 0} (-1)^i \mathrm{Tr}(F^*,H^i(X,\mathbb{Q}_{\ell}))$$ where the LHS $X^F = X_0(\mathbb{F}_q)$, is the number of fixed point of $F$ on $X$, which is again the number of rational points of $X_0$. 

Last, we recall the celebrated Riemann hypothesis

• (Riemann hypothesis over finite fields) Let $X_0/\mathbb{F}_q$ be a variety, then the eigenvalues of $F^* \colon H^i(X,\mathbb{Q}_{\ell}) \longrightarrow H^i(X,\mathbb{Q}_{\ell})$ are algebraic integers of the form $\omega q^{j/2}$ with $0 \leq j \leq i$. When $X_0$ is smooth, we know that $j=i$.
  

Some applications of $\ell$-adic cohomology

Despite a long list of beatiful properties, we still lack some pieces to study $\ell$-adic cohomology of the Drinfeld curve. That is, the group action; equivalently, we'd like some equivariant $\ell$-adic cohomology.

Let $X_0/\mathbb{F}_q$ be a quasi-projective variety (may this reduce to Cech cohomology) and $M$ be a monoid acting on a variety $X_0$ by endormorphisms (then $M$ also acts on $X$ by endormorphism). The following is the content of  R. Rouquier, Complexes de chaînes étales et courbes de Deligne-Lusztig

Proposition 2(Rouquier). There exists a complex $R\Gamma_c(X,\mathbb{Q}_{\ell})$ (one can replace $\mathbb{Q}_{\ell}$ with $\mathbb{Z}_{\ell}$ or $\mathbb{Z}/\ell^n\mathbb{Z}$) of $\mathbb{Q}_{\ell}[M]$-modules whose cohomology groups $H^i_c(X,\mathbb{Q}_{\ell})$ are $\mathbb{Q}_{\ell}[M]$-modules.

Now all the properties listed above admit their equivariant counterparts when one puts the action at the right place. There are two new properties that need to be clarified in my opinions:

Lemma 3. If $X_0/\mathbb{F}_q$ is a variety on which a finite group $G$ acts, then $H^i_c(X/G,\mathbb{Q}_{\ell}) \simeq H^i_c(X,\mathbb{Q}_{\ell})^G$ as $\mathbb{Q}_{\ell}[G]$-modules. 

Proof. First, quasi-projectivity here is important for the existence of $X/G$. Indeed, in general, a finite group $G$ acts on $X$ admits a quotient $X/G$ iff every orbit of $G$ is contained in an affine open set and for quasi-projective variety, one has this property (every finite set of points is contaned in an affine open subscheme). 

Second, this property does not hold when one works with "smaller" coefficients like $\mathbb{Z}_{\ell}$ or $\mathbb{Z}/\ell^n\mathbb{Z}$. It only holds for characteristic zero coefficients like $\mathbb{Q}_{\ell}$ or $\overline{\mathbb{Q}}_{\ell}$. The proof below explains this phenomenon. 

The Hochschild-Serre spectral sequence implies that there are spectral sequence for each $n \geq 0$

$$H^i(G,H^j_c(X,\mathbb{Z}/\ell^n\mathbb{Z})) \Rightarrow H^{i+j}_c(X/G,\mathbb{Z}/\ell^n\mathbb{Z})$$ so we are done if this spectral sequence degenerates for $i > 0$ because $$H^0(G,H^j_c(X,\mathbb{Z}/\ell^n\mathbb{Z})) = H^j_c(X,\mathbb{Z}/\ell^n\mathbb{Z})^G.$$ However, with torsion coefficients, this is NOT true. Luckily, groups (with $i>0$) on the LHS are killed by $\left| G \right|$ (this is a fun exercise to show that group cohomology $H^i(G,M)$ are killed by $\left|G \right|$). Therefore, $H^{i>0}(G,H^j_c(X,\mathbb{Z}_{\ell}))$ is also killed by $\left|G \right|$ and hence it must be zero after we pass to $\mathbb{Q}_{\ell}$, showing the degeneration of the spectral with coefficients $\mathbb{Q}_{\ell}$. 

Lemma 4. Let $k$ be an algebraically closed field and $G/k$ be a connected algebraic group acting on a $k$-variety $X$, then the action of $G$ on $H^i_c(X,\mathbb{Z}/n\mathbb{Z})$ are trivial. Consequently, the same conclusion holds for $H^i_c(X,\mathbb{Q}_{\ell})$. 

Proof.  Let $\pi \colon G \times X \longrightarrow X$ be the projection onto the second factor and $f \colon G \times X \longrightarrow G \times X$ be the automorphism defined by $(g,x) \longmapsto (g,gx)$. Note that $\pi \circ f = \pi$. Consider the cartesian square $$\require{AMScd} \begin{CD} G \times X @>{\pi}>> X;\\ @VVV @VVV \\ G @>>> \operatorname{Spec}(k); \end{CD}$$ and use proper base change, we obtain

$$R^i\pi_!(\mathbb{Z}/n\mathbb{Z}) \simeq \underline{H^i_c(X,\mathbb{Z}/n\mathbb{Z})}$$ as constant sheaves on $G$. The automorphism $f^*$ acts on this sheaf and at $g \in G$, it acts the way $g$ acts on $H^i_c(X,\mathbb{Z}/n\mathbb{Z})$. At the identity, the action of $f$ is trivial. An endomorphism of a constant sheaf over a connected base is constant, making the action of $f$ trivial everywhere. 

Finally, we record some (hard!) supplementary results that are consequences of trace formula.

Theorem 5. If $g$ is an automorphism of $X/\overline{\mathbb{F}}_q$ of finite order (say, of order $n$) and $X$ is defined by $X_0/\mathbb{F}_q$, then 

$$\mathrm{Tr}(g^* \mid H^{\bullet}_c(X,\mathbb{Q}_{\ell})) = \sum_{i \geq0}(-1)^i\mathrm{Tr}(g^* \mid H^i_c(X,\mathbb{Q}_{\ell})) = \lim_{t \to \infty} \sum_{n=1}^{\infty} \left| X^{gF^n} \right| t^n$$ is an integer independent of $\ell$. 

Proof. Let us prove the equality first. By base change to $\mathbb{F}_{q^n}$, the morphism $gF^n$ is a Frobenious morphism of $X_1/\mathbb{F}_{q^n}$ with $X_1= X_0 \otimes_{\mathbb{F}_q}\mathbb{F}_{q^n}$. By definition, $g$ commutes with $F$ then an elementary step of linear algebra shows that we can simultaneously triangonalize $g^*,F^*$ on each $H^i_c(X,\mathbb{Q}_{\ell})$. Let $\alpha_{ij},\beta_{ij}$ be eigenvalues of $F^*,g^*$. Thus, one finds that

$$\mathrm{Tr}(gF^n \mid H^{\bullet}_c(X,\mathbb{Q}_{\ell})) =  \sum \pm \beta_{ij}\alpha_{ij}^n.$$ Hence, 

$$\sum_{n=1}^{\infty} \left| X^{gF^n} \right| t^n = \sum \pm \beta_{ij} \sum (\alpha_{ij}t)^n  = \sum \pm \beta_{ij}\left( \frac{1}{1 - \alpha_{ij}t} - 1\right).$$ Let $t$ approach $\infty$, the RHS becomes $\sum \pm \beta_{ij}$, which is $\mathrm{Tr}(g^* \mid H^{\bullet}_c(X,\mathbb{Q}_{\ell}))$. 

Now let us prove that this is an integer independent of $\ell$. The first part and the Riemann hypothesis combine showing that $\mathrm{Tr}(g^* \mid H^{\bullet}_c(X,\mathbb{Q}_{\ell}))$ is an algebraic integer. Therefore, it remains to show that it is a rational number independent of $\ell$. Independence of $\ell$ is clear from the identity. For each $n \geq 1$, $\left |X^{gF^n} \right|$ is a rational. Let $\varphi$ be any automorphism of $\overline{\mathbb{Q}}_{\ell}$. For $t$ rationa, we have

$$\varphi \left(  \sum_{n=1}^{\infty} \left| X^{gF^n} \right| t^n\right ) = \sum_{n=1}^{\infty} \left| X^{gF^n} \right| t^n$$ and so $\sum_{n=1}^{\infty} \left| X^{gF^n} \right| t^n$ is a rational. Taking limit, we obtain

$$\varphi \left( \lim_{t \to \infty} \left(  \sum_{n=1}^{\infty} \left| X^{gF^n} \right| t^n\right )  \right)= \lim_{t \to \infty} \sum_{n=1}^{\infty} \left| X^{gF^n} \right| t^n$$ (this step makes sense due to the presentation above in terms of $\alpha_{ij},\beta_{ij}$) which shows that $\mathrm{Tr}(g^* \mid H^{\bullet}_c(X,\mathbb{Q}_{\ell}))$ is fixed by any automorphism of $\overline{\mathbb{Q}}_{\ell}$, thus a rational. 

Remark 6. If $X_0/\mathbb{F}_q$ is smooth, then the above is even true for any endomorphism of $X$ thanks to a result of Katz and Messing.

Remark 7. Here some subtle points that are about to occur. Let $T$ be a torus acting on a scheme $X$. The space of fixed points $X^T$ is NOT a scheme a priori but an algebraic space. Fortunately, the classical theory of algebraic spaces (with Bialynicki-Birula decomposition as the core stone) with $\mathbb{G}_m$-actions shows that $X^T$ is indeed a scheme (but to do so, one needs to view $X,X^T$ as algebraic spaces first and algebraicity of the ground field is important). 

Otherwise, one can view $X^T$ as algebraic space and does their etale cohomology since the theory is well-developed even for algebraic stacks.

Theorem 8 (Degline + Lusztig). Let $X_0/\mathbb{F}_q$ be a variety and $g$ is an automorphism of $X$ of finite order such that $g = uv = vu$ with order of $u$ prime to $p$ and order of $v$ a power of $p$, then 

$$\mathrm{Tr}(g^* \mid H^{\bullet}_c(X,\mathbb{Q}_{\ell})) = \mathrm{Tr}(v^* \mid H^{\bullet}_c(X^u,\mathbb{Q}_{\ell}))$$ holds. 

Proof. Too long, did not read.

Proposition 9 (Deligne + Michel). Let $X_0/\mathbb{F}_q$ be a variety and $T/\overline{\mathbb{F}}_q$ be a torus acting on $X$. Let $g$ be an automorphism of $X$ of finite order commuting with the action of $T$, then 

$$\mathrm{Tr}(g^* \mid H^{\bullet}_c(X,\mathbb{Q}_{\ell}))  = \mathrm{Tr}(g^* \mid H^{\bullet}_c(X^t,\mathbb{Q}_{\ell}))$$ for any $t \in T$. Moreover, if $X$ affine, then 

$$\mathrm{Tr}(g^* \mid H^{\bullet}_c(X,\mathbb{Q}_{\ell}))  = \mathrm{Tr}(g^* \mid H^{\bullet}_c(X^T,\mathbb{Q}_{\ell})).$$ Proof. We decompose $g = uv = vu$ as in Deligne, Lusztig Theorem 8. Then by this theorem, 

$$\mathrm{Tr}(g^* \mid H^{\bullet}_c(X,\mathbb{Q}_{\ell}) = \mathrm{Tr}(v^* \mid H^{\bullet}_c(X^u,\mathbb{Q}_{\ell})).$$ The action of $T$ commutes with $g$, so $T$ acts on $X^u$. As $T$ is connected, Lemma 4 points out that the action of any $t \in T$ on $H^{\bullet}_c(X^u,\mathbb{Q}_{\ell})$ is trivial. We find that

$$\begin{align*} \mathrm{Tr}(v^* \mid H^{\bullet}_c(X^u,\mathbb{Q}_{\ell})) & = \mathrm{Tr}((vt)^* \mid H^{\bullet}_c(X^u,\mathbb{Q}_{\ell})) \\ & =  \mathrm{Tr}(v^* \mid H^{\bullet}_c(X^{ut},\mathbb{Q}_{\ell})) \\ & = \mathrm{Tr}(v^* \mid H^{\bullet}_c(X^{tu},\mathbb{Q}_{\ell})) \\ & = \mathrm{Tr}(g^* \mid H^{\bullet}_c(X^t,\mathbb{Q}_{\ell}) \end{align*}$$ thanks to Deligne, Lusztig Theorem 8. If $X$ is affine, then there exists some $t \in T$ so that $X^T  = X^t$ (non-trivial, omitted).

Corollary 10. Let $X_0/\mathbb{F}_q$ be an affine variety and $T/\overline{\mathbb{F}}_q$ be a torus acting on $X$. Let $G$ be a finite group. If $G \times T$ acts on $X$ so that the action of $G,T$ commute, then

$$\mathrm{Tr}(g^* \mid H^{\bullet}_c(X,\mathbb{Q}_{\ell})) = \mathrm{Tr}(g^* \mid H^{\bullet}_c(X^T,\mathbb{Q}_{\ell}))$$ for any $g \in G$. 

Remark 11. In the appendix $A$ of book Representations of $SL_2(\mathbb{F}_q)$ of Cédric Bonnafé , I believe that Theorem A.2.6(e) is incorrectly stated, the correct statement should be the one replacing $H^{\bullet}$ with traces of elements in $\Gamma$.  

Remark 12. Every thing in fact holds for varieties over arbitrary algebraically closed fields, but varieties comes from finite fields are all we need.  Finally, every result above is in fact true after replacing $\mathbb{Q}_{\ell}$ by any of its algebraic extension; in particular, $\overline{\mathbb{Q}}_{\ell}$. For a finite extension $E/\mathbb{Q}_{\ell}$, one can mimic the formula in the case $\mathbb{Q}_{\ell}$. Let $\mathcal{O} \subset E$ be the ring of integers with a fixed uniformizer $\pi$, then define

$$H^i(X,E)  = \left(\varprojlim H^i(X,\mathcal{O}/\pi^n\mathcal{O})  \right) \otimes_{\mathcal{O}} E$$ and to get coefficients in $\overline{\mathbb{Q}}_{\ell}$, we take the limits of those $H^i(X,E)$.

Drinfeld curves

Over the finite field $\mathbb{F}_q$, the Drinfeld curve is the affine curve

$$C = \operatorname{Spec}\left(\mathbb{F}_q[x,y]/(xy^q - x^qy - 1) \right)$$ and there are several actions on this curve, which are what making it interesting to study and allowing us to compute its cohomology. 

• The action of $SL_2(\mathbb{F}_q)$ on $C$ via multiplication of matrices and positive characteristic miracle $$\begin{pmatrix}  ax + by & (ax+by)^q  \\ cx+dy & (cx+dy)^q \\ \end{pmatrix} = \begin{pmatrix} a & b \\  c & d \\ \end{pmatrix}\begin{pmatrix} x & x^q \\ y & y^q \\ \end{pmatrix}.$$  
• The action of roots of unity $\mu_{q+1}$ by homotheties $$\zeta \cdot (x,y) = (\zeta x, \zeta y).$$
• The Frobenious endomorphism $$F \colon C \longrightarrow C, (x,y) \mapsto (x^q,y^q).$$ This morphism is $SL_2(\mathbb{F}_q)$-equivariant but not $\mu_{q+1}$-equivariant. 

Lemma 12. The curve $C$ is affine, smooth, irreducible.

Proof. Affineness is obvious. Smoothness follows from fact that the differential is $(y^q, -x^q)$. Irreducibility follows from the irreducibility of the polynomial $f(x,y) = xy^q - x^qy-1$. Indeed, by a change of variables, $(z,t) = (x/y,1/y)$, we have to show that $g(z,t) = t^{q+1}-z^q - z$ irreducible in $\mathbb{F}_q[z][t]$. We note that $z^q - z \in (z)$ but not in $(z^2)$. Thus, one can apply Eisenstein's criterion with the domain $\mathbb{F}_q[z]$ and a prime ideal $(z)$. 

We have

• (Artin vanishing) If $X_0/\mathbb{F}_q$ is smooth, affine (both are essential), then $H^i_c(X,\mathbb{Q}_{\ell})=0$ for all $0 \leq i < \dim(X)$. This strengthens the finiteness theorem. 

Corollary 13. By Poincare duality, $H^0_c(C_{\overline{\mathbb{F}}_q},\mathbb{Q}_{\ell}) = H^2(C_{\overline{\mathbb{F}}_q},\mathbb{Q}_{\ell})=0$ and $H^2_c(C_{\overline{\mathbb{F}}_q},\mathbb{Q}_{\ell}) = \mathbb{Q}_{\ell}$.

Although we can not determine $H^1_c(C)$, we are still capable to describe its equivariant parts. This is done via its quotients. We need a technical criterion for this purpose.

Proposition 14. Let $k$ be a field and $X,Y$ be two smooth, irreducible varieties. Let $\varphi \colon X \longrightarrow Y$ be a $k$-morphism and $G$ be a finite group acting on $X$. Assume that:

• $\varphi$ is surjective.
• $\varphi(x) = \varphi(x')$ iff $x,x'$ are in the same $G$-orbit.
• There exists $x_0 \in X$ such that the differential of $\varphi$ at $x_0$ is surjective.

Then there is an induced isomorphism $X/G \overset{\sim}{\longrightarrow} Y$. 

Although I've never read the proof of this, I guess the idea (intuitively) is that the third condition implies that $\varphi$ is like a submersion and plus smoothness, $\varphi$ looks like a projection $\varphi(\mathbf{x},\mathbf{y}) = \mathbf{x}$. However, the second condition tolds us that $\mathbf{y} = \varnothing$, which show that $\varphi$ is an isomorphism.

Proposition 15. We endow $\mathbb{A}^1$ with the action of $\mu_{q+1}$ via $\zeta \cdot x = \zeta^2 x$. The morphism $\gamma \colon C \longrightarrow \mathbb{A}_{\mathbb{F}_q}^1$ given at the level of points by $(x,y) \longmapsto (xy^{q^2} - x^{q^2}y)$ is a $(\mu_{q+1},F)$-equivariant isomorphism. 

Proof. Proven by Proposition 14, details omitted.

Proposition 16. We endow $\mathbb{G}_m$ with the action of $\mu_{q+1}$ via $\zeta \cdot x = \zeta x$. The morphism $\gamma \colon C \longrightarrow \mathbb{G}_m$ given at the level of points by $(x,y) \longmapsto y$ is a $(\mu_{q+1},F)$-equivariant isomorphism.

Proof.  Proven by Proposition 14, details omitted.

Proposition 17. We endow $\mathbb{P}^1$ with the natural action of $SL_2(\mathbb{F}_q)$ and the action of Frobenius $[x:y] \longmapsto [x^q :y^q]$. The morphism $C \longrightarrow \mathbb{P}^1 \setminus \mathbb{P}^1(\mathbb{F}_q)$ given at the level of points by $(x,y) \longmapsto [x:y]$ is a $(SL_2(\mathbb{F}_q),F)$-equivariant isomorphism.

Proof.  Proven by Proposition 14, details omitted.

Geometric Mackey formula 

Let us fix an isomorphism $\overline{\mathbb{Q}}_{\ell} \simeq \mathbb{C}$. For every character $\theta \colon \mu_{q+1} \longrightarrow \overline{\mathbb{Q}}^{\times}_{\ell} = \mathbb{C^{\times}}$ and a complex representation $\mu_{q+1} \longrightarrow \mathrm{GL}(V)$, we denote

$$V[\theta] \overset{\text{def}}{=} \left \{v \in V \mid \zeta \cdot v = \theta(\zeta)v \ \forall \zeta \in \mu_{q+1} \right \}.$$ For a finite group $G$ and a complex representation $G \longrightarrow \mathrm{GL}(V)$, we implicitly fix an inner product $\left<-,- \right>$. The goal of the blog today is the following

Theorem (geometric Mackey, INCORRECTLY STATED). Given characters $\theta_1,\theta_2$ of $\mu_{q+1}$, then

$$\left< H^1_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\theta_1],H^1_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\theta_2] \right>_{SL_2(\mathbb{F}_q)} = \left<\theta_1,\theta_2 \right>_{\mu_{q+1}} + \left<\theta_1,\theta_2^{-1}\right>_{\mu_{q+1}}.$$ Why do I say that this's incorrectly stated (which is the one found in Shurui Liu's note)? One has to go to the end to understand this. Let's analyze the LHS.

Lemma 18.  For the Drinfeld curve and any character $\theta$ of $\mu_{q+1}$, one has $$H^1_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\theta] = H^1_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\theta^{-1}] = (H^1_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\theta])^{\vee}.$$

Proof. One notes that for the Frobenius $F \colon C \longrightarrow C$ we have $F(\zeta x,\zeta y) = \zeta^{-1}F(x,y)$ for any $\zeta \in \mu_{q+1}$. Moreover, $F$ is $SL_2(\mathbb{F}_q)$-equivariant. Thus, after base change to $\overline{\mathbb{F}}_q$, $F$ induces an $SL_2(\mathbb{F}_q)$-equivariant isomorphism

$$H^1_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell}) \longrightarrow H^1_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})$$ which sends $H^1_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\theta]$ isomorphically onto $H^1_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\theta^{-1}]$. The second equality is obvious.

Now the LHS in the geometric Mackey formula becomes

$$\begin{align*} \left< H^1_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\theta_1],H^1_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\theta_2] \right>_{SL_2(\mathbb{F}_q)} &  = \left< (H^1_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\theta_1])^{\vee},H^1_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\theta_2] \right>_{SL_2(\mathbb{F}_q)}\\ & = \left< 1,H^1_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\theta_1] \otimes H^1_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\theta_2] \right>_{SL_2(\mathbb{F}_q)} \\ & = \dim_{\mathbb{C}}\left( H^1_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\theta_1] \otimes H^1_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\theta_2]  \right)^{SL_2(\mathbb{F}_q)}\\ & = \dim_{\mathbb{C}}\left( H^{\bullet}_c(C \times C,\overline{\mathbb{Q}}_{\ell})[\theta_1 \times \theta_2]\right)^{SL_2(\mathbb{F}_q)}. \end{align*}$$ Now the ideal is clear thanks to Kunneth formula, we should understand $H^{\bullet}_c(C \times C, \overline{\mathbb{Q}}_{\ell})$. We'll cut $C \times C$ into pieces that we understand. We set $Z = C \times C \subset \mathbb{A}^4_{\mathbb{F}_q} = \operatorname{Spec}(\mathbb{F}_q[x,y,z,t])$. Let $Z = Z_0 \cup Z_{\neq 0}$ with $Z_0$ cut out by $xw - yz = 0$ and $Z_{\neq 0}$ the complement. Hence,

$$\begin{align*} \dim_{\mathbb{C}}\left( H^{\bullet}_c(C \times C,\overline{\mathbb{Q}}_{\ell})[\theta_1 \times \theta_2]\right)^{SL_2(\mathbb{F}_q)} & =  \dim_{\mathbb{C}}\left( H^{\bullet}_c(Z_0,\overline{\mathbb{Q}}_{\ell})[\theta_1 \times \theta_2]\right)^{SL_2(\mathbb{F}_q)} + \dim_{\mathbb{C}}\left( H^{\bullet}_c(Z_{\neq 0},\overline{\mathbb{Q}}_{\ell})[\theta_1 \times \theta_2]\right)^{SL_2(\mathbb{F}_q)} \\ & = \dim_{\mathbb{C}}\left( H^{\bullet}_c(Z_0/SL_2(\mathbb{F}_q),\overline{\mathbb{Q}}_{\ell})[\theta_1 \times \theta_2]\right) + \dim_{\mathbb{C}}\left( H^{\bullet}_c(Z_{\neq 0}/SL_2(\mathbb{F}_q),\overline{\mathbb{Q}}_{\ell})[\theta_1 \times \theta_2]\right) \end{align*}$$ thanks to Mayer-Vietoris and Lemma, provided that these quotients exist, as shown below.

Lemma 19. The inclusion $Z_0 \subset Z$ is $\mu_{q+1} \times \mu_{q+1} \times SL_2(\mathbb{F}_q)$-equivariant. In other words, the subvariety $Z_0 \subset Z$ is stable under the action of $\mu_{q+1} \times \mu_{q+1} \times SL_2(\mathbb{F}_q)$. 

Proof. Obvious. 

Proposition 20. The morphism $\mu_{q+1} \times C \longrightarrow Z_0$ given at the level of points by $(\zeta,x,y) \longmapsto (x,y,\zeta x,\zeta y)$ is $SL_2(\mathbb{F}_q)$-equivariant and induces an $\mu_{q+1}\times \mu_{q+1}$-equivariant isomorphism $Z_0/SL_2(\mathbb{F}_q) \simeq \mu_{q+1} \times \mathbb{A}^1$, where the RHS is acted on by $\mu_{q+1}\times \mu_{q+1}$ via $(\zeta_1,\zeta_2)\cdot (\zeta,z) = (\zeta_1^{-1}\zeta_2\zeta, \zeta_1^2 z)$. 

Proof. This is done by hands.

Corollary 21. With notation as above, one has

$$H^{\bullet}_c(Z_0/SL_2(\mathbb{F}_q),\overline{\mathbb{Q}}_{\ell}) =  H^{\bullet}_c(\mu_{q+1} \times \mathbb{A}^1,\overline{\mathbb{Q}}_{\ell}) = H^{\bullet}_c(\mu_{q+1}) \otimes H^{\bullet}_c(\mathbb{A}^1) = H^{\bullet}(\mu_{q+1}).$$ If we view $\mu_{q+1}^{(2)} = \mu_{q+1} \subset \mu_{q+1} \times \mu_{q+1}$ via $\zeta \longmapsto (\zeta,\zeta^{-1})$, then 

$$H^{\bullet}_c(Z_0/SL_2(\mathbb{F}_q) \simeq H^{\bullet}_c(\mu_{q+1}) = \mathrm{Ind}_{\mu_{q+1}^{(2)}}^{\mu_{q+1} \times \mu_{q+1}}1$$ as representations.

Proposition 22. Let $V \subset \mathbb{G}_m \times \mathbb{A}^2$ be the subvariety cut out by $w^{q+1} - xy = 0$, where $w$ is the coordiate on $\mathbb{G}_m$ and $x,y$ are coordinates on $\mathbb{A}^2$. The map 

$$Z_{\neq 0} \longrightarrow \mathbb{G}_m \times \mathbb{A}^2$$ given by

$$(x,y,z,t) \longmapsto (xt - yz, xt^q - yz^q,x^qt - y^qz)$$ induces an isomorphism $Z_{\neq 0}/SL_2(\mathbb{F}_q) \simeq V$. Moreover, there is an $\mu_{q+1}\times \mu_{q+1}$-equivariant isomorphism $V^{\mathrm{G}_m} = \mu_{q+1}$ where the action on $\mu_{q+1}$ is given by $(\zeta_1,\zeta_2)\cdot \zeta = \zeta_1\zeta_2\zeta$.

Proof.  Use Proposition 14, details omitted. The calculation is quite painful but direct, see page 42 of Cédric Bonnafé book.

Corollary 23. With notation as above, one DOES NOT has

$$H^{\bullet}_c(Z_{\neq 0}/SL_2(\mathbb{F}_q),\overline{\mathbb{Q}}_{\ell}) =  H^{\bullet}_c(V,\overline{\mathbb{Q}}_{\ell}) \overset{\text{WRONG}}{=} H^{\bullet}_c(V^{\mathbb{G}_m},\overline{\mathbb{Q}}_{\ell}) = H^{\bullet}_c(\mu_{q+1},\overline{\mathbb{Q}}_{\ell}).$$ However, one has 

$$\mathrm{Tr}(g^* \mid H^{\bullet}_c(Z_{\neq 0}/SL_2(\mathbb{F}_q),\overline{\mathbb{Q}}_{\ell})) = \mathrm{Tr}(g^* \mid H^{\bullet}_c(\mu_{q+1},\overline{\mathbb{Q}}_{\ell}))$$ for any  $g \in \mu_{q+1} \times \mu_{q+1}$. If we view $\mu_{q+1}^{(1)} = \mu_{q+1} \subset \mu_{q+1} \times \mu_{q+1}$ via $\zeta \longmapsto (\zeta,\zeta)$, then 

$$H^{\bullet}_c(\mu_{q+1},\overline{\mathbb{Q}}_{\ell}) = \mathrm{Ind}_{\mu_{q+1}^{(1)}}^{\mu_{q+1} \times \mu_{q+1}}$$ as representations. 

The reason behind the middle equality comes from the statement of Corollary, which holds only for traces and not cohomology. To correct the statement of the geometric Mackey formula, one has to introduce the Degline-Lusztig induction, associated with a character $\theta

$$R_C(\theta) =  H^0_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\theta] - H^1_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\theta] + H^2_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\theta] = - H^1_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\theta] + H^2_c(C_{\overline{\mathbb{F}}_q},\overline{\mathbb{Q}}_{\ell})[\theta]$$

which is not a "real" representation, but a "virtual" one, i.e., living in some Grothendieck rings of representations. Now we can state the correct version of geometric Mackey. 

Theorem 24 (geometric Mackey). For characters $\theta_1,\theta_2$ of $\mu_{q+1}$, one has

$$\left< R_C(\theta_1),R_C(\theta_2) \right>_{SL_2(\mathbb{F}_q)} = \left<\theta_1,\theta_2 \right>_{\mu_{q+1}} + \left<\theta_1,\theta_2^{-1}\right>_{\mu_{q+1}}.$$ Here comes to the point that one has to use the full power of results in the last part on $\ell$-adic cohomology. Details I prefer to omit. Now the rest is easy 

$$\left< R_C(\theta_1),R_C(\theta_2) \right>_{SL_2(\mathbb{F}_q)}  = \dim \mathrm{Ind}_{\mu_{q+1}^{(1)}}^{\mu_{q+1} \times \mu_{q+1}}1[\theta_1 \times \theta_2] + \dim \mathrm{Ind}_{\mu_{q+1}^{(2)}}^{\mu_{q+1} \times \mu_{q+1}}1[\theta_1 \times \theta_2] = \left< \theta_1,\theta_2 \right> + \left<\theta_1,\theta_2^{-1} \right>$$ by definition.