id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-5245	Alabbad, Fatimah	Decomposition of the Unitary Representation of $SL_{2}(\mathbb{R})$ on the Upper Half Plane into Irreducible Components	2024	14	.pdf	application/pdf	4492	282	81	It is less known that there are related actions of SL2(R) on dual and double numbers which have the form z = x+ ιy, ι2 = 0 or ι2 = 1, correspondingly. A generic cycle [9], § 4.2 is the set of points (u, v) ∈ R2 defined for all values of σ by the equation k(u2 − σv2)− 2lu− 2nv +m = 0. (1) This equation is represented by a point (k, l, n,m) from a projective space P3, since for a scaling factor λ ̸= 0, the point (λk, λl, λn, λm) defines an equation equivalent to (1).	cache/ejpam-5245.pdf	txt/ejpam-5245.txt