2025/06/04 by Chan, Charlotte, Oi, Masao · 3 citations
#FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2506.04449
Under a largeness assumption on the size of the residue field, we give an explicit description of the positive-depth Deligne--Lusztig induction of unramified elliptic pairs (T,θ). When θ is regular, we show that positive-depth Deligne--Lusztig induction gives a geometric realization of Kaletha's Howe-unramified regular L-packets. This is obtained as an immediate corollary of a very simple "litmus test" characterization theorem which we foresee will have interesting future applications to small-p constructions. We next define and analyze Green functions of two different origins: Yu's construction (algebra) and positive-depth Deligne--Lusztig induction (geometry). Using this, we deduce a comparison result for arbitrary θ from the regular setting. As a further application of our comparison isomorphism, we prove the positive-depth Springer hypothesis in the 0-toral setting and use it to give a geometric explanation for the appearance of orbital integrals in supercuspidal character formulae.