Broad-band infrared (IR) spectroscopy, especially at high spectral resolution, is a largely unexplored area for the far IR (FIR) and submm wavelength region due to the lack of proper grating technology to produce high resolution within the very constrained volume and weight required for space mission instruments. High resolution FIR spectroscopy is an essential tool to resolve many atomic and molecular lines to measure physical and chemical conditions and processes in the environments where galaxy, star and planets form. A silicon immersion grating (SIG), due to its over three times high dispersion over a traditional reflective grating, offers a compact and low cost design of new generation IR high resolution spectrographs for space missions. A prototype SIG high resolution spectrograph, called Florida IR Silicon immersion grating spectromeTer (FIRST), has been developed at UF and was commissioned at a 2 meter robotic telescope at Fairborn Observatory in Arizona. The SIG with 54.74 degree blaze angle, 16.1 l/mm groove density, and 50x86 mm2 grating area has produced R=50,000 in FIRST. The 1.4-1.8 um wavelength region is completely covered in a single exposure with a 2kx2k H2RG IR array. The on-sky performance meets the science requirements for ground-based high resolution spectroscopy. Further studies show that this kind of SIG spectrometer with an airborne 2m class telescope such as SOFIA can offer highly sensitive spectroscopy with R~20,000-30,000 at 20 to 55 microns. Details about the on-sky measurement performance of the FIRST prototype SIG spectrometer and its predicted performance with the SOFIA 2.4m telescope are introduced.