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Zhiyu
Zhiyu
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Let $k=\Bbb F_q$ be a finite field, $G$ be a connected reductive group over $k$, $\mathcal{N}$ be the nilpotent cone of $G$ which consists of nilpotent elements in the lie algebra of $G$.

Motivation & example: If $G=GL_n$, then $\#\mathcal{N}(k)=q^{n^2-n}$ (here is a direct proof) i.e there are exactly $q^{\text{dim}\mathcal{N}}$ nilpotent matrices over $k$.

The direct proof uses no geometry while the result is simple, so I am interested in an explanation by Grothendieck trace formula. However, $\mathcal{N}$ is not smooth ( $\mathcal{N} \cong \{xy=z^2\}$ if $G={GL}_2$ ) in general, so interesection cohomology might be better than etale cohomology (The nilpotent cone is rationally smooth when using intersection cohomology at least in characterestic zero).

Q1. What is $\#\mathcal{N}(k)$ for general $G$?

Q2 What is the cohomology (etale/intersection cohomology) of $\mathcal{N}_{\bar k}$ and the trace of geometric Frobenius action on it?

Q3 Do we have Grothendieck trace formula for intersection cohomology? (

My vague idea for Q1 is to consider Springer resolution and reduce to counting size of the flag variety and Springer fiber.

Let $k=\Bbb F_q$ be a finite field, $G$ be a connected reductive group over $k$, $\mathcal{N}$ be the nilpotent cone of $G$ which consists of nilpotent elements in the lie algebra of $G$.

Motivation & example: If $G=GL_n$, then $\#\mathcal{N}(k)=q^{n^2-n}$ (here is a direct proof) i.e there are exactly $q^{\text{dim}\mathcal{N}}$ nilpotent matrices over $k$.

The direct proof uses no geometry while the result is simple, so I am interested in an explanation by Grothendieck trace formula. However, $\mathcal{N}$ is not smooth ( $\mathcal{N} \cong \{xy=z^2\}$ if $G={GL}_2$ ) in general, so interesection cohomology might be better than etale cohomology (The nilpotent cone is rationally smooth when using intersection cohomology at least in characterestic zero).

Q1. What is $\#\mathcal{N}(k)$ for general $G$?

Q2 What is the cohomology (etale/intersection cohomology) of $\mathcal{N}_{\bar k}$ and the trace of geometric Frobenius action on it?

Q3 Do we have Grothendieck trace formula for intersection cohomology? (

My vague idea for Q1 is to consider Springer resolution and reduce to counting size of the flag variety and Springer fiber.

Let $k=\Bbb F_q$ be a finite field, $G$ be a connected reductive group over $k$, $\mathcal{N}$ be the nilpotent cone of $G$ which consists of nilpotent elements in the lie algebra of $G$.

Motivation & example: If $G=GL_n$, then $\#\mathcal{N}(k)=q^{n^2-n}$ (here is a direct proof) i.e there are exactly $q^{\text{dim}\mathcal{N}}$ nilpotent matrices over $k$.

The direct proof uses no geometry while the result is simple, so I am interested in an explanation by Grothendieck trace formula. However, $\mathcal{N}$ is not smooth ( $\mathcal{N} \cong \{xy=z^2\}$ if $G={GL}_2$ ) in general, so interesection cohomology might be better than etale cohomology (The nilpotent cone is rationally smooth when using intersection cohomology at least in characterestic zero).

Q1. What is $\#\mathcal{N}(k)$ for general $G$?

Q2 What is the cohomology (etale/intersection cohomology) of $\mathcal{N}_{\bar k}$ and the trace of geometric Frobenius action on it?

Q3 Do we have Grothendieck trace formula for intersection cohomology?

My vague idea for Q1 is to consider Springer resolution and reduce to counting size of the flag variety and Springer fiber.

Source Link
Zhiyu
Zhiyu
  • 6.6k
  • 2
  • 11
  • 44

Number of points of the nilpotent cone over a finite field and its cohomology

Let $k=\Bbb F_q$ be a finite field, $G$ be a connected reductive group over $k$, $\mathcal{N}$ be the nilpotent cone of $G$ which consists of nilpotent elements in the lie algebra of $G$.

Motivation & example: If $G=GL_n$, then $\#\mathcal{N}(k)=q^{n^2-n}$ (here is a direct proof) i.e there are exactly $q^{\text{dim}\mathcal{N}}$ nilpotent matrices over $k$.

The direct proof uses no geometry while the result is simple, so I am interested in an explanation by Grothendieck trace formula. However, $\mathcal{N}$ is not smooth ( $\mathcal{N} \cong \{xy=z^2\}$ if $G={GL}_2$ ) in general, so interesection cohomology might be better than etale cohomology (The nilpotent cone is rationally smooth when using intersection cohomology at least in characterestic zero).

Q1. What is $\#\mathcal{N}(k)$ for general $G$?

Q2 What is the cohomology (etale/intersection cohomology) of $\mathcal{N}_{\bar k}$ and the trace of geometric Frobenius action on it?

Q3 Do we have Grothendieck trace formula for intersection cohomology? (

My vague idea for Q1 is to consider Springer resolution and reduce to counting size of the flag variety and Springer fiber.