Return to Question

Let $G$ be a reductive group over a finite field $k$, let $F$ be the Frobenius morphism on $G$.

I'll start with a somewhat vague question and make my question more specific further down:

How do cuspidal representations fit into Deligne-Lusztig characters $R_{T,\theta}$?

A little about what is known classically: if $\rho$ is a cuspidal representation of $G^F$, then $\rho$ is a subrepresentation of $\pm R_{T,\theta}$ for some pair $(T,\theta)$, where $T$ is a minisotropic torus (so it is not a subtorus of any proper Levi subgroup of $G$). Moreover, for such a $T$, if $\theta: T^F \to \mathbb{C}^\times$ is in general position (so not fixed by any nontrivial element of the Weyl group), then $\pm R_{T,\theta}$ is irreducible cuspidal.

In the case of $G = GL_n$, it turns out that these are all of the irreducible cuspidal representations; i.e. every irreducible cuspidal representation is isomorphic to some $\pm R_{T,\theta}$, where $T$ is ministropic, and $R_{T,\theta} \cong R_{T',\theta'}$ if and only if $(T,\theta)$ is $G^F$-conjugate to $(T',\theta')$.

Of course, it's too much to hope that this should happen in every case. Let $T$ be an anisotropic torus in $SL_2$. Then there is a character $\theta: T^F \to \mathbb{C}^\times$ that is not in general position, where $\pm R_{T,\theta}$ splits as a sum of two irreducible cuspidal representations (perhaps you need the base field to have odd characteristic).

Now to specify my question. Throughout, $T$ is a minisotropic torus of $G$, and $\theta:T^F \to \mathbb{C}^\times$.

1). Are there groups $G$ with pairs $(T,\theta)$ and cuspidal representations $\rho,\, \rho'$ such that $\rho$ occurs in $R_{T,\theta}$ with positive multiplicity, and $\rho'$ occurs with negative multiplicity?

2). Are there groups $G$ with pairs $(T,\theta)$ such that $R_{T,\theta}$ such that $R_{T,\theta}$ contains both cuspidal and non-cuspidal representations with nonzero multiplicity?

Let $G$ be a reductive group over a finite field $k$, let $F$ be a Frobenius morphism on $G$.

I'll start with a somewhat vague question and make my question more specific further down:

How do cuspidal representations fit into Deligne-Lusztig characters $R_{T,\theta}$?

A little about what is known classically: if $\rho$ is a cuspidal representation of $G^F$, then $\rho$ is a subrepresentation of $\pm R_{T,\theta}$ for some pair $(T,\theta)$, where $T$ is a minisotropic torus (so it is not a subtorus of any proper Levi subgroup of $G$). Moreover, for such a $T$, if $\theta: T^F \to \mathbb{C}^\times$ is in general position (so not fixed by any nontrivial element of the Weyl group), then $\pm R_{T,\theta}$ is irreducible cuspidal.

In the case of $G = GL_n$, it turns out that these are all of the irreducible cuspidal representations; i.e. every irreducible cuspidal representation is isomorphic to some $\pm R_{T,\theta}$, where $T$ is ministropic, and $R_{T,\theta} \cong R_{T',\theta'}$ if and only if $(T,\theta)$ is $G^F$-conjugate to $(T',\theta')$.

Of course, it's too much to hope that this should happen in every case. Let $T$ be an anisotropic torus in $SL_2$. Then there is a character $\theta: T^F \to \mathbb{C}^\times$ that is not in general position, where $\pm R_{T,\theta}$ splits as a sum of two irreducible cuspidal representations (perhaps you need the base field to have odd characteristic).

Now to specify my question. Throughout, $T$ is a minisotropic torus of $G$, and $\theta:T^F \to \mathbb{C}^\times$.

1). Are there groups $G$ with pairs $(T,\theta)$ and cuspidal representations $\rho,\, \rho'$ such that $\rho$ occurs in $R_{T,\theta}$ with positive multiplicity, and $\rho'$ occurs with negative multiplicity?

2). Are there groups $G$ with pairs $(T,\theta)$ such that $R_{T,\theta}$ such that $R_{T,\theta}$ contains both cuspidal and non-cuspidal representations with nonzero multiplicity?

Let $G$ be a reductive group over a finite field $k$, let $F \in G_k$ be a Frobenius element, so that $G^F$ is a finite group of Lie type.

I'll start with a somewhat vague question and make my question more specific further down:

How do cuspidal representations fit into Deligne-Lusztig characters $R_{T,\theta}$?

A little about what is known classically: if $\rho$ is a cuspidal representation of $G^F$, then $\rho$ is a subrepresentation of $\pm R_{T,\theta}$ for some pair $(T,\theta)$, where $T$ is a minisotropic torus (so it is not a subtorus of any proper Levi subgroup of $G$). Moreover, for such a $T$, if $\theta: T^F \to \mathbb{C}^\times$ is in general position (so not fixed by any nontrivial element of the Weyl group), then $\pm R_{T,\theta}$ is irreducible cuspidal.

In the case of $G = GL_n$, it turns out that these are all of the irreducible cuspidal representations; i.e. every irreducible cuspidal representation is isomorphic to some $\pm R_{T,\theta}$, where $T$ is ministropic, and $R_{T,\theta} \cong R_{T',\theta'}$ if and only if $(T,\theta)$ is $G^F$-conjugate to $(T',\theta')$.

Of course, it's too much to hope that this should happen in every case. Let $T$ be an anisotropic torus in $SL_2$. Then there is a character $\theta: T^F \to \mathbb{C}^\times$ that is not in general position, where $\pm R_{T,\theta}$ splits as a sum of two irreducible cuspidal representations (perhaps you need the base field to have odd characteristic).

Now to specify my question. Throughout, $T$ is a minisotropic torus of $G$, and $\theta:T^F \to \mathbb{C}^\times$.

1). Are there groups $G$ with pairs $(T,\theta)$ and cuspidal representations $\rho,\, \rho'$ such that $\rho$ occurs in $R_{T,\theta}$ with positive multiplicity, and $\rho'$ occurs with negative multiplicity?

2). Are there groups $G$ with pairs $(T,\theta)$ such that $R_{T,\theta}$ such that $R_{T,\theta}$ contains both cuspidal and non-cuspidal representations with nonzero multiplicity?

Let $G$ be a reductive group over a finite field $k$, let $F$ be the Frobenius morphism on $G$.

I'll start with a somewhat vague question and make my question more specific further down:

How do cuspidal representations fit into Deligne-Lusztig characters $R_{T,\theta}$?

A little about what is known classically: if $\rho$ is a cuspidal representation of $G^F$, then $\rho$ is a subrepresentation of $\pm R_{T,\theta}$ for some pair $(T,\theta)$, where $T$ is a minisotropic torus (so it is not a subtorus of any proper Levi subgroup of $G$). Moreover, for such a $T$, if $\theta: T^F \to \mathbb{C}^\times$ is in general position (so not fixed by any nontrivial element of the Weyl group), then $\pm R_{T,\theta}$ is irreducible cuspidal.

In the case of $G = GL_n$, it turns out that these are all of the irreducible cuspidal representations; i.e. every irreducible cuspidal representation is isomorphic to some $\pm R_{T,\theta}$, where $T$ is ministropic, and $R_{T,\theta} \cong R_{T',\theta'}$ if and only if $(T,\theta)$ is $G^F$-conjugate to $(T',\theta')$.

Of course, it's too much to hope that this should happen in every case. Let $T$ be an anisotropic torus in $SL_2$. Then there is a character $\theta: T^F \to \mathbb{C}^\times$ that is not in general position, where $\pm R_{T,\theta}$ splits as a sum of two irreducible cuspidal representations (perhaps you need the base field to have odd characteristic).

Now to specify my question. Throughout, $T$ is a minisotropic torus of $G$, and $\theta:T^F \to \mathbb{C}^\times$.

1). Are there groups $G$ with pairs $(T,\theta)$ and cuspidal representations $\rho,\, \rho'$ such that $\rho$ occurs in $R_{T,\theta}$ with positive multiplicity, and $\rho'$ occurs with negative multiplicity?

2). Are there groups $G$ with pairs $(T,\theta)$ such that $R_{T,\theta}$ such that $R_{T,\theta}$ contains both cuspidal and non-cuspidal representations with nonzero multiplicity?