Language: Persian ?
Source: Period account

Description

Karan shāhī is the name of a specific kind of Indian hilt, commonly used for the talwar sword. 

Karan shahi hilt as illustrated in Hendley's Damascening on steel or iron as practiced in India. 1892.

Characteristics

The karan shahi style is quite distinct with pointed grip section and wide quillons. As a type, it is associated primarily with the Marwar region in general and Jodhpur in particular.

Hermen Goetz, writing in 1950, sheds some light on the origin of the term:

"The costly damascened swords in pure Mughal taste (figs. 67, 68), were first introduced by Karan Singhjī and, therefore, were termed "Karan Shāhī".

Other hilt types

In addition, Hendley makes mention of a hakim khāni and hakim shahi hilts.

Two talwar hilts illustrated in Hendley's Damascening on steel or iron as practiced in India. 1892.¹
Left: Hakim khāni
Right: Hakim shahi

It is never explicitly mentioned in the literature, but going by the illustrations in Hendley's work we can gather that the general rule is as follows:

Hakim khāni hilts have a smooth transition between grip section and crossbar. 
Hakim shahi hilts have a pronounced V shape where the grip merges into the crossbar.

Notes
1. Hermann Goetz; The Art and Architecture of Bikaner State. Bruno Cassirer, Oxford. 1950. Page 124 - 125
2. Thomas Holbein Hendley; Damascening on steel or iron as practiced in India. W. Griggs & Sons, Ltd. London 1892.