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$\mathrm{GL}(n,\mathbb{Z}_p)$-invariant Gaussian measures on the space of $p$-adic polynomials

Yassine EL Maazouz, Antonio Lerario

TL;DR

This work develops a nonarchimedean analogue of Kostlan’s invariant-measures classification by studying $\mathrm{GL}(n,\mathbb{Z}_p)$-invariant gaussian distributions on the space of homogeneous polynomials $\mathbb{Q}_p[x_1,\dots,x_n]_{(d)}$. It translates gaussian measures to invariant lattices via Evans’ framework and analyzes these lattices inside Bruhat–Tits buildings, proving that, under suitable core conditions, invariant lattices (and hence invariant gaussian measures) are finite up to scaling and often unique when the partition is $p$-core (e.g., $p>d$). The irreducibility of Schur representations for $\ell$-core partitions and the fixed-point results in the building underpin both the lattice- and measure-theoretic conclusions, yielding a precise nonarchimedean counterpart to Kostlan’s real/complex classifications. Additionally, the paper develops an integral-geometry framework for the expected number of roots of random invariant polynomial systems and connects it to volumes determined by $R$-structures. Together, these results provide a rigorous foundation for probabilistic algebraic geometry over nonarchimedean fields and offer concrete tools for studying random polynomial systems in the $p$-adic setting.

Abstract

We prove that if $p>d$ there is a unique gaussian distribution (in the sense of Evans) on the space $\mathbb{Q}_p[x_1, \ldots, x_n]_{(d)}$ which is invariant under the action of $\mathrm{GL}(n, \mathbb{Z}_p)$ by change of variables. This gives the nonarchimedean counterpart of Kostlan's Theorem on the classification of orthogonally (respectively unitarily) invariant gaussian measures on the space $\mathbb{R}[x_1, \ldots, x_n]_{(d)}$ (respectively $\mathbb{C}[x_1, \ldots, x_n]_{(d)}$). More generally, if $V$ is an $n$--dimensional vector space over a nonarchimedean local field $K$ with ring of integers $R$, and if $λ$ is a partition of an integer $d$, we study the problem of determining the invariant lattices in the Schur module $S_λ(V)$ under the action of the group $\mathrm{GL}(n,R)$.

$\mathrm{GL}(n,\mathbb{Z}_p)$-invariant Gaussian measures on the space of $p$-adic polynomials

TL;DR

This work develops a nonarchimedean analogue of Kostlan’s invariant-measures classification by studying -invariant gaussian distributions on the space of homogeneous polynomials . It translates gaussian measures to invariant lattices via Evans’ framework and analyzes these lattices inside Bruhat–Tits buildings, proving that, under suitable core conditions, invariant lattices (and hence invariant gaussian measures) are finite up to scaling and often unique when the partition is -core (e.g., ). The irreducibility of Schur representations for -core partitions and the fixed-point results in the building underpin both the lattice- and measure-theoretic conclusions, yielding a precise nonarchimedean counterpart to Kostlan’s real/complex classifications. Additionally, the paper develops an integral-geometry framework for the expected number of roots of random invariant polynomial systems and connects it to volumes determined by -structures. Together, these results provide a rigorous foundation for probabilistic algebraic geometry over nonarchimedean fields and offer concrete tools for studying random polynomial systems in the -adic setting.

Abstract

We prove that if there is a unique gaussian distribution (in the sense of Evans) on the space which is invariant under the action of by change of variables. This gives the nonarchimedean counterpart of Kostlan's Theorem on the classification of orthogonally (respectively unitarily) invariant gaussian measures on the space (respectively ). More generally, if is an --dimensional vector space over a nonarchimedean local field with ring of integers , and if is a partition of an integer , we study the problem of determining the invariant lattices in the Schur module under the action of the group .
Paper Structure (26 sections, 14 theorems, 88 equations, 1 figure)

This paper contains 26 sections, 14 theorems, 88 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Probabilistic Algebraic Geometry.
  3. Invariant lattices.
  4. Group actions on the Bruhat-Tits Buildings.
  5. Structure of the paper
  6. Probabilistic motivation
  7. Gaussian distributions on real vector spaces.
  8. Invariant gaussian distributions on the space of real and complex polynomials.
  9. Gaussian distributions on -adic spaces.
  10. Invariant gaussian distributions on the space of adic polynomials.
  11. Background and preliminaries
  12. The group .
  13. The Schur functor.
  14. Lattices.
  15. Some auxiliary results from representation theory.
  16. ...and 11 more sections

Key Result

Theorem 1.1

There are finitely many gaussian distributions on $\mathbb{Q}_p[x_1, \ldots, x_n]_{(d)}$ (up to multiples) which are invariant under the action of $\mathop{\mathrm{GL}}\nolimits(n, \mathbb{Z}_p)$ by change of variables. Moreover, if $p>d$, there is only one such distribution (up to multiples), which

Figures (1)

  • Figure 1: A convex set in the building $\mathcal{B}_2(\mathbb{Q}_2)$ (the set of vertices colored in blue).

Theorems & Definitions (36)

  • Theorem 1.1
  • Example 1.2: $n=2, d=2$
  • Proposition 1.3
  • Theorem 1.4
  • Remark 3.1
  • Example 3.2
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 26 more