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Abstract

There are many formulas that express interesting properties of a finite group
G in terms of sums over its characters. For estimating these sums, one of the
most salient quantities to understand is the character ratio
trace(\pi(g)) / dim(\pi), for an irreducible representation \pi of G and an
element g of G. It turns out [Gurevich-Howe15, Gurevich-Howe17] that for
classical groups G over finite fields there are several (compatible) invariants
of representations that provide strong information on the character ratios. We
call these invariants collectively rank. Rank suggests a new way to organize
the representations of classical groups over finite and local fields - a way in
which the building blocks are the "smallest" representations. This is in
contrast to Harish-Chandra's philosophy of cusp forms that is the main
organizational principle since the 60s, and in it the building blocks are the
cuspidal representations which are, in some sense, the "LARGEST". The
philosophy of cusp forms is well adapted to establishing the Plancherel formula
for reductive groups over local fields, and led to Lusztig's classification of
the irreducible representations of such groups over finite fields. However,
analysis of character ratios might benefit from a different approach. In this
note we discuss further the notion of tensor rank for GL_n over a finite field
F_q and demonstrate how to get information on representations of a given tensor
rank using tools coming from the recently studied eta correspondence, as well
as the well known philosophy of cusp forms, mentioned just above. A significant
discovery so far is that although the dimensions of the irreducible
representations of a given tensor rank vary by quite a lot (they can differ by
large powers of q), for certain group elements of interest the character ratios
of these irreps are nearly equal to each other.