We prove dimension bounds on the jet schemes of the variety of nilpotent matrices (and of related varieties) in positive characteristic. This result has applications to the analytic properties of the Chevalley map that sends a matrix to its characteristic polynomial. We show that our dimension bound implies, under the assumption of existence of resolution of singularities in positive characteristic, that the Chevalley map pushes a smooth compactly supported measure to a measure whose density function is $L^t$ for any $t<\infty$. We also prove this analytic property of the Chevalley map, unconditionally, when the characteristic of the field exceeds $n/2$. The zero characteristic counterpart of this result is an important step in the proof of the celebrated Harish-Chandra's integrability theorem. In a sequel work [AGKSb], we show that also in positive characteristic, this analytic statement implies Harish-Chandra's integrability theorem for cuspidal representations of the general linear group.
Results on dimensions of jet schemes. Fix a finite field F ℓ . Unless explicitly stated otherwise, all the algebraic varieties that we consider will be defined over F ℓ . For a variety X we denote by J m (X) its m-th jet scheme. We consider J m as a functor from the category of varieties to the category of schemes. We fix an integer n and set g := gl n considered as an algebraic variety.
In this paper we prove:
Theorem A ( §8). Let N ⊂ g be the nilpotent cone. There is a constant C 0 such that for any m ∈ N we have
We deduce from this result bounds on jet schemes of more varieties. To formulate these bounds we make: Notation 1.1.1. Denote by • c -the affine space of monic polynomials of degree n. We will identify it with A n . • p : g → c -the Chevalley map (essentially sending an element to its characteristic polynomial).
• For an integer i ∈ N we denote by g i := g × c . . . × c g i times the i-folded fiber product of g with itself over c with respect to the map p.1
We deduce the following:
Theorem B ( §8). There is a constant C such that for any x ∈ c and any m ∈ N we have
From this we deduce the following:
Theorem C ( §8). For any i there is a constant C i such that for any m ∈ N we have dim J m (g i ) < m dim(g i ) + C i .
1.2. Results on pushforward of measures. We deduce from the results above the following one.
Theorem D ( §11). Let i ∈ N. Assume that the variety g i admits a strong resolution of singularities. Let µ c be a Haar measure on c. Then for any smooth compactly supported measure µ on g := g(F ), there exists a function
such that p * (µ) = f µ c .
Remark 1.2.1. In §12 we give several alternative conditions on resolution of singularities under which the result holds.
Finally we show that one can replace the assumption of Theorem D on the existence of resolution with an assumption on characteristic:
Theorem E ( §13). Suppose char(F ℓ ) > n 2 . Let F := F ℓ ((t)). Then for any smooth compactly supported measure µ on g := g(F ), the measure p * (µ) can be written as a product of a function in L ∞ (c) and a Haar measure on c.
1.3.1. FRS maps. Theorems D and E are related to the notion of FRS maps introduced and studied in [AA16]. Let us recall this notion: Definition 1.3.1. A map ϕ : X → Y of smooth algebraic varieties over a field of characteristic zero is called FRS if it is flat, its fibers are reduced, and the singularities of its fibers are rational.
The motivation to this definition is the following: Theorem 1.3.2 ([AA16, Theorem 3.4], [Rei18]). Let ϕ be a map of smooth algebraic varieties over a local field F of characteristic zero.
If ϕ is FRS then for any smooth compactly supported measure µ on X(F ), the measure ϕ * (µ) on Y := Y(F ) can be written as a product of a continuous function and a smooth measure on Y.
Unfortunately, we do not have an extension of this Theorem to the positive characteristic case. In fact it is not even clear how to formulate it correctly since there is no universally accepted definition of rational singularities (see [Har98,Smi97,Bha12,Kov00] for several related notions).
For this paper we choose the following notion of rational singularities in positive characteristic.
Definition 1.3.3. Let Z be a variety defined over an arbitrary field. We say that the singularities of Z are rational if Z is Cohen-Macaulay, normal, and admits a resolution of singularities η : Z → Z such that the natural morphism η * (Ω Z) → i * (Ω Z sm ) is an isomorphism. Here i : Z sm → Z is the embedding of the smooth locus and Ω denotes the sheaf of top differential forms.
Remark 1.3.4. In characteristic zero, this notion is equivalent to rational singularities, see e.g. [AA16, Appendix B, Proposition 6.2].
Next we give several extensions of the notion of FRS maps to positive characteristic: Definition 1.3.5. Let ϕ : M → N be a flat morphism of smooth algebraic varieties over a local field F of arbitrary characteristic. Assume that the fibers of ϕ are reduced and normal.
(1) We say that ϕ is geometrically FRS (in short geo-FRS) if for any y ∈ N( F ), the singularities of ϕ -1 (y) are rational.
(2) We say that ϕ is analytically FRS (in short an-FRS) if for any smooth compactly supported measure µ M on M := M(F ) there exist a smooth compactly supported measure µ N on N := N(F ), and a bounded function f on N such that ϕ * (µ M ) = f µ N .
(3) We say that ϕ is almost analytically FRS (in short almost an-FRS) if for any smooth compactly supported measure µ M on M there exist smooth compactly supported measure µ N and a function f on N such that ϕ * (µ M ) = f µ N . and f ∈ L r (N ) for all r ∈ [1, ∞) Using extension of scalars from F ℓ to F ℓ ((t)) we will apply these notions also for maps of varieties over F ℓ .
Remark 1.3.6. As in Remark 1.3.4, in characteristic zero, the geo-FRS property is equivalent to FRS property. Also, by Theorem 1.3.2, in this case each of them implies the an-FRS property.
In this language, Theorem D implies that, under an appropriate a
This content is AI-processed based on open access ArXiv data.