Model transition for certain Langlands quotients of GL _{4 <i>n</i>}

Abstract In this paper, we study model transitions for the Langlands quotient of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>GL</m:mi> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>F</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{GL}_{4n}(F)} corresponding to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>σ</m:mi> <m:mo stretchy="false">|</m:mo> <m:mo>⋅</m:mo> <m:msup> <m:mo stretchy="false">|</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:msup> <m:mo>×</m:mo> <m:msup> <m:mi>σ</m:mi> <m:mo>∨</m:mo> </m:msup> <m:mo stretchy="false">|</m:mo> <m:mo>⋅</m:mo> <m:msup> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:mrow> </m:msup> </m:mrow> </m:math> {\sigma|\cdot|^{\frac{1}{2}}\times\sigma^{\vee}|\cdot|^{-\frac{1}{2}}} with an irreducible tempered representation σ of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>GL</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>F</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{GL}_{2n}(F)} when F is p -adic field. In particular, we explicitly construct an isomorphism between <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>Sp</m:mi> <m:mrow> <m:mn>4</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>F</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{Sp}_{4n}(F)} -distinguished models and certain degenerate Whittaker models for the above Langlands quotient.