In this article, we introduce a partial Poisson transform on the affine symmetric space SU(n,n)/SL(n,C)xR^*₊ and prove that this transform is a continuous SU(n,n)-homomorphism. We also give the form for the Fourier transform of the Poisson kernel.

We would like to thank Professor Heping Liu for helpful conversations during the preparation of this paper. The second author was partially supported by the National Natural Science Foundation of China (Grant Nos.12161083 and 12126360) and the Natural Science Foundation of Xinjiang Uygur Autonomous Region (Grant Nos. 2019D01C049 and 2020D01C048). Most of this work was completed when the second author was visiting Tsinghua University and Nanjing Normal University. He would like to thank the Department of Mathematical Sciences of Tsinghua University and School of Mathematical Sciences and Mathematical Institute of Nanjing Normal University for warm hospitality. He would also like to thank Professors Bin Xu and Huicheng Yin for their support and encouragement during this work.