Spherical Shalika models on PGU _2,2 and the theta correspondence for (PGSp ₄ , PGU _2,2 )

Abstract We study Shalika models for generic unramified representations of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>PGU</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:math> {{\rm PGU}_{2,2}} over non-archimedean local fields of characteristic zero. We show that they are unique up to constant by means of the theta correspondence for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>PGSp</m:mi> <m:mn>4</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>PGU</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(\mathrm{P}{{\rm GSp}}_{4},{\rm PGU}_{2,2})} . We then prove a Casselman–Shalika formula which relates the values of spherical Shalika functionals on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>PGU</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:math> {{\rm PGU}_{2,2}} to the values of finite-dimensional complex representations of the dual group of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>PGSp</m:mi> <m:mn>4</m:mn> </m:msub> </m:math> {\mathrm{P}{{\rm GSp}}_{4}} .