Closest point to a dual lattice point (in particular for root lattices!)

Question

Given a lattice $\Lambda\subset \mathbb{R}^n$ and a point $p\in\mathbb{R}^n$ outside the lattice, then I known it is a hard question to determine the set $S\subset \Lambda$ of all lattice points with minimal distance (NP-hard in some formulation, right?)

I wonder what is known in case $p\in\Lambda^*$ is in the dual lattices. More specifically for every class $[p]\in\Lambda^*/\Lambda$. I would like to know (or know about) the number of such closest points.

AND I wonder what is known in case $\Lambda$ is a root lattice or a rescaled root lattice ?

Example: For the 2-dimensional triangular lattice i.e. root lattice $A_2$, I checked that apart from $[p]=[0]$, which has by definition always $1$ closest point namely $p$ itself, the two other cosets $[p]=[\lambda_1],[\lambda_2]$ each have $3$ closest points (together the $6$ minimal vectors of the dual lattice with normsquare 2/3)

Thanx for any helpful comments!

Simon

PS: If you must know...the multiplicities should count the groundstates of the conformal quantum field theory associated to a lattice and in particular the dimensions of the irreducible representations of Zhu's algebra, which I would like to work with...

Answer 1

For a general $\Lambda$ the assumption $p \in \Lambda^*$ doesn't help, because $\Lambda$ can be rescaled to make any $p$ arbitrarily close to a dual lattice point.

The special case of a scaled root lattice is easy, because you can just subtract roots from a coset representative until it can be made no smaller. But again this does not use the hypothesis $p \in \Lambda^*$.

If $\Lambda$ is an (unscaled) root lattice then the minimal norms of the cosets of $\Lambda$ in $\Lambda^*$ have structural significance; for instance they arise as corrections to the canonical heights of sections of an elliptic surface. Explicitly they are as follows:

$\bullet$ The trivial coset of course has minimal norm zero always.

$\bullet$ in $A_{n-1}$, the coset consisting of zero-sum vectors with all coordinates in ${\bf Z} + \frac m n$ $(m=1,2,\ldots,n-1)$ has minimal norm $m(n-m) / n$; there are $n \choose m$ minimal representatives, with $m$ entries of $(m-n)/n$ and the remaining $n-m$ entries all equal $m/n$.

$\bullet$ $D_n$ has three nontrivial cosets, one of minimal norm $1$ and the other two of minimal norm $n/4$. Minimal norm $1$ is attained by $\pm e_m$ (the $2n$ signed unit vectors), and $n/4$ by $2^{n-1}$ vectors in each coset with every coordinate equal $\pm 1/2$. Note that this is consistent with $D_3 \cong A_3$ and with the triality automorphisms of $D_4$.

$\bullet$ Finally, $E_6$ has two nontrivial cosets which are each other's negatives and have minimal norm $4/3$ realized by $27$ vectors in each coset (the $27$ lines of a cubic); there is only one minimal coset in $E_7$, of minimal norm $3/2$ realized by $56$ minimal vectors (whose $28$ $\pm$-pairs correspond to the bitangents of a quartic curve); and $E_8$ is its own dual so has no nontrivial cosets at all.