Let $G$ be a split semisimple algebraic group scheme over $\mathbf{Z}$ (I'm mostly interested in the case $G = Sp_4$).

Let $V$ be an irreducible representation of the generic fibre $G_{\mathbf{Q}}$, of some highest weight $\lambda$, and $v$ a choice of highest-weight vector. Then there is a notion of an "admissible $\mathbf{Z}$-lattice" in $V$: it's a $\mathbf{Z}$-lattice $V_{\mathbf{Z}}$ in $V$ for which the structure map $G_{\mathbf{Q}} \to GL(V)$ extends to a map of $\mathbf{Z}$-group schemes $G_\mathbf{Z} \to GL(V_\mathbf{Z})$, and such that the intersection of $V_{\mathbf{Z}}$ with the highest-weight space $\mathbf{Q} \cdot v$ is $\mathbf{Z} \cdot v$. There are finitely many of these, and in particular there's a maximal and a minimal one.

My question is this: let $P$ be a parabolic subgroup (containing our fixed choice of Borel) and $M$ its Levi factor. It's easy to see that $V$ breaks up as a direct sum of eigenspaces uder $Z(M)$, and the highest-weight eigenspace is the weight $\lambda$ representation $W$ of $M$, with $v$ as its highest weight vector. Moreover, the image in $W$ of an admissible lattice in $V$ is an admissible lattice in $W$.

Is the projection to $W$ of the maximal (or minimal) admissible lattice in $V$ equal to the maximal (resp. minimal) admissible lattice in $W$?

(If this works for the minimal lattice it works for the maximal one, and vice versa, because the minimal and maximal lattices are dual to each other.)

  • $\begingroup$ Why are there only finitely many of "these"? I assume you work up to conjugation by some group, but which group? $GL(V_{\mathbb Z})(\mathbb Z)$ or $GL(V_{\mathbb Z})(\mathbb Q)$? Or some other group? (I know this is not what your question is about, but your statement sparked my curiosity.) $\endgroup$ – Ariyan Javanpeykar Nov 10 '16 at 13:44
  • $\begingroup$ Once you fix the highest-weight vector $v$ there are literally only finitely many admissible $V_{\mathbf{Z}}$'s -- up to conjugation by the trivial group! $\endgroup$ – David Loeffler Nov 10 '16 at 16:48
  • $\begingroup$ Aha! Thank you for your comment. :) $\endgroup$ – Ariyan Javanpeykar Nov 10 '16 at 17:41

Let $\mathfrak{n}_-$ be the Lie algebra of the derived subgroup $N_-$ of $B_{-}$ where $B_{-}=TN_-$ is a Borel subgroup of $G$ opposite to a Borel subgroup $B_+\subset P$ (here $T$ is a split maximal torus of $G$ defined over $\mathbb{Z}$). If $V_\mathbb{Z}$ is a minimal admissible $\mathbb{Z}$-lattice then there is a highest weight vector $v\in V$ (for $B_+$) such that $V_{\mathbb{Z}}=U^{-}_{\mathbb Z}\cdot v$ where $U^{-}_{\mathbb Z}$ is the Kostant $\mathbb{Z}$-form of the universal enveloping algebra $U(\mathfrak{n}_-)$. It is spanned over $\mathbb{Z}$ by the PBW-monomials in $e_\alpha^n/n!$ where $\alpha$ is a negative root of $G$ with respect to $T$ and $n\in \mathbb{Z}_{\ge 0}$ (the root vectors $e_\alpha$ should come from a Chevalley basis of $\mathfrak{g}={\rm Lie}(G)$). It follows that $V_\mathbb{Z}\cap W$ is spanned over $\mathbb{Z}$ by the elements of the form $X\cdot v$ where $X$ is a PBW monomial in $e_\alpha^n/n!$ and $\alpha$ is a negative root of $P$ with respect to $T$. This shows that $V_\mathbb{Z}\cap W$ is a minimal admissible lattice for a Levi subgroup of $P$.

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  • $\begingroup$ Great! Do you have any suggestions for where I might find references for some of the statements you use here? $\endgroup$ – David Loeffler Nov 9 '16 at 13:05
  • $\begingroup$ Most of what I used can be found in ``Lectures on Chevalley groups '' by Robert Steinberg. The book is easy to find on the web and it can be downloaded from several sources. $\endgroup$ – Alexander Premet Nov 9 '16 at 13:19

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